Mathematics

Probability Theory: 7 Complete Quick Facts

October 6, 2023January 25, 2021 by DR. MOHAMMED MAZHAR UL HAQUE

Probability theory emerged from the concept of taking risk. there are many complication today that come from the game of chance, such as wining a football match, playing cards and throwing a coin or throwing a dice.

Probability theory is used in many different sectors and the pliability of probability theory furnishes tools for almost so many different requirements. Here we are going to discuss the probability theory and few samples with the help of some fundamental concepts and results.

RANDOM EXPERIMENTS:

“Random experiment is a kind of experiments where the result cannot be predicted.”

SAMPLE SPACE:

The set of all possible outcomes from the experiment is called the sample space, it is usually denoted by S and all test out comes are said to be a sample point.
Eg: Think about the random Experiment of tossing 2 coins at a time. There are 4 outcomes constitute a sample space denoted by, S ={ HH, TT, HT, TH}

TRAIL & EVENT:

Each non-empty subset of A of the sample space S is called an event. Consider the experiment to throw a coin. When we throw a coin, we can find a head (H) or a tail (T). Here throwing a coin is the trail and getting a head or a tail is a an event.

COMPOUND EVENTS:

Events acquired by combining two or more basic events are called compound events or decomposable events.

EXHAUSTIVE EVENTS:

The total number of feasible results of any trail is called exhaustive events.

Eg: In throwing a dice the potential results are 1 or 2 or 3 or 4 or 5 or 6. So we have a total of 6 events in throwing die.

MUTUALLY EXCLUSIVE AND EXHAUSTIVE SYSTEM OF EVENTS :

Let S is sample space of random experiment, If X₁, X₂, …..X_n are the subsets of S and

(i) X_i∩ X_j =Φ for i ≠ j and (ii) X₁ ∪ X₂ ………∪ X_n =S

Then this collection of X₁∪ X₂ ………∪ X_n is said to create a mutually exclusive and exhaustive system of events.

What is Independence?

When we pull out a card in a pocket of well-adjusted cards and secondly we also extract a card from the rest packet of cards (containing 51 cards), then the second extracting hangs on the first. But if, on the other hand, we pull the second card out of the pack by inserting the first card drawn(replacing), the second draw is known as independent of the first.

Example: Two coins are thrown. Let the first coin having head be event X and the Y be the second coin showing tail after throw. Two events X and Y are basically independent.

Example: Two fair dice are drawn. If odd number come on first die consider it as event X and for second die even number as event Y.

The two event X and Y are mutually independent.

Example: A card is drawn from a pack of 52 cards. If A = card is of Hearts, B = card is an King and A ⋂ B = card is King of Hearts, then events A and B are dependent

FAVOURABLE NUMBER OF CASES: The number of cases which permit an event to be tried in a trial is the total number of primary events that the aspect of any of them ensures the occurrence of the event.

What is meant by Probability

If a arbitrary demonstration results in n incongruous , equally likely and exhaustive outcomes, out of which m are agreeable to the occurrence of an event A, then the probability of happening of A is given by

Probability notation: P(X)=m/n

For two events X and Y,

(i) X′ or x̄ or X^C indicates for the non-occurrence or negation of X.

(ii) X ∪ Y means for the occurring of at least any one of X and Y.

(iii) X ∩ Y means for the concurrent occurrence of X and Y.

(iv) X′ ∩ Y′ means for the non-occurrence of one and the other X and Y.

(v) X⊆ Y means for “the happening of X indicates occurrence of Y”.

Example: A bucket contains 6 red & 7 black marbles. Find the probability of drawing a red color marbles.

Solution: Total no. of possible ways of getting 1 marble= 6 + 7

Number of ways of getting 1 red marble= 6

Probability = (Number of favorable cases)/(Total number of exhaustive cases) = 6/13

Example: From a pack of 52 cards, 1 card is drawn at random. Find the probability of getting a queen card.

Solution: A queen card may be chosen in 4 ways.

Total number of ways of selecting 1 queen card = 52

Probability = Number of favorable cases/Total number of exhaustive cases = 4/52=1/13

Example: Find the probability of throwing:

(a) getting 4 , (b) an odd number, (c) an even number

with an ordinary die (six faced).

Solution: The problem is dice problem

a) When throwing a die there is only one way of getting 4.

Probability = Number of favorable cases/Total number of exhaustive cases = 1/6

b) Number of ways of falling an odd number is 1, 3, 5 = 3

Probability = Number of favorable cases/Total number of exhaustive cases = 3/6=1/2

c) Number of ways of falling an even number is 2, 4, 6 = 3

Probability = Number of favorable cases/Total number of exhaustive cases = 3/6=1/2

Example: What is the possible chance of finding a king and a queen, when 2 cards are drawn from a pack of 52 playing cards?

Solution: 2 cards can be drawn from a pack of 52 cards = ⁵²C₂ (52 choose 2) ways

⁵²C₂=52!/2!(52-2)!=(52*51)/2=1326

1 queen card can be picked from 4 queen cards = ⁴C₁=4 ways (4 choose 1)

1 king card can be taken from 4 king cards = ⁴C₁=4 ways (4 choose 1)

Favorable cases = 4 × 4 = 16 ways

P(drawing 1 queen & 1 king card) = Number of favorable cases/Total number of exhaustive cases = 16/1326=8/663

Example: What are the chances of getting 4, 5 or 6 in the first throw and 1, 2, 3 or 4 in the second throw if the dice are thrown twice.

Solution:

Let P(A) = probability of getting 4, 5 or 6 in the first throw=3/6=1/2

and P(B)= probability of getting 1, 2, 3 or 4 in the second throw= 4/6=2/3

be the probability of the events then

Example: A book having total 100 number of pages, if any one of the page is selected arbitrary. What is the possible chance that the sum of all the digits of the page number of the selected page is 11.

Solution: The number of Favorable ways to get 11 will be (2, 9), (9, 2), (3, 8), (8, 3), (4, 7), (7, 4), (5, 6), (6, 5)

Hence required probability = 8/100=2/25

Example: A bucket contains 10 white, 6 red, 4 black & 7 blue marbles. 5 marbles are pull out at random. What is the probability that 2 of them are red colour and one is black colour ?

Solution:

Total no. of marbles= 10 + 6 + 4 + 7 =27

5 marbles can be drawn from these 27 marbles= 27 choose 5 ways

= ²⁷C₅=27!/

$5!(27-5)!$

=(27*26*25*24*23)/(5*4*3*2)=80730

Total no. of exhaustive events = 80730

2 red marbles can be drawn from 6 red marbles= 6 ways

= ⁶C₂=6!/

$2!(6-2)!$

=(6*5)/2=15

1 black marbles can be pull out from 4 black marbles= 4 choose 1 ways= ⁴C₁=4

∴ No. of favorable cases = 15 × 4 = 60

Hence required probability = Number of favourable casesTotal number of exhaustive cases

Conclusion:

The probability theory is very interesting and applicable in our daily day to day life so probability theory and examples looks familiar to us, this is actually a complete theory which is used now a days in numerous technologies and applications, This article was just a glimpse of the concept of probability the consecutive articles will deal with the detail concept and results of Probability, for more study, please refer below book:

Ref: Schaum’s Outlines of Probability and Statistics.

If you are interested in reading other topics on mathematics please see this page.

Characteristics of Function Graphs: 5 Important Facts

December 31, 2023January 23, 2021 by TheMachineScience Core SME

Characteristics of Function Graphs

Characteristics of Function Graphs, this article will discuss the concept of graphical presentation of functions in addition to the value of a variable present in a function. So that the readers can easily understand the methodology.

Which graph represents the functions f(X) = |x-2| – 1 ?

One look at the right hand side expression makes us wonder, what are those two bars around -2 ? Well those bars are the notation for a very special function in mathematics, known as the modulus function or the absolute value function. This function is so important in function theory that it is worth a few words on its origin.

Let us say we are to decide the time required to go from one city to another. In this case, won’t we only be interested in the distance between the two cities? Will the direction be of any importance? Similarly, in the study of calculus, we are often required to analyze the closeness of two numbers, which is the absolute value of their difference. We don’t care if the difference is positive or negative. German mathematician Karl Weierstrass was the one who realized the necessity of a function which would express the absolute value of a number. In the year 1841, Weierstrass defined the Modulus function and used the two bars as its symbol.

f(x) = x for all x>0

=-x for all x<0

= 0 for x=0

Abbreviated as f(x) = |x|

From the definition, it is clear that this function does not have any effect on a positive number. It however changes a negative number to a positive number having the same absolute value. Hence

|5| = 5

7-2 = 5

|-5| = 5

|2-7| = 5

To draw the graph of |x|, we should start with the graph of f(x) = x which simply is a straight line through the origin, inclined at 45 degrees to the positive side of the X axis

Characteristics of Function Graphs: **Function Theory : f(x) = x**

It can be said that the upper half of this graph will be retained by f(x) = |x| as this function doesn’t change positive numbers. The lower half of the graph, however, has to change side, because |x| must always be positive. So, all the points on the lower half of f(x)=x will now be replaced on the upper half, keeping the same distance from the X axis. In other words, the entire LEFT HALF OF f(x) = |x| IS ACTUALLY THE REFLECTION OF THE LOWER HALF OF f(x) = x about the X axis.

Characteristics of Function Graphs: **Function theory:** |x| and x graphs

In the above figure, the right half shows the graphs of |x| and x superimposed, while the left half shows one as the reflection of another. It is essential to note that this technique may be stretched to any function. In other words, it is easy to imagine the graph of |f(x)| if we already know the graph of f(x). Replacing the lower half with its reflection about the X axis is the key.

Now we know how to plot |x|. But our original problem demands the plot of |x-2|. Well, this is nothing but a shift of origin from (0,0) to (2,0) as it simply decreases the X reading of all points by 2 units, thus transforming f(x) into f(x-2).

Characteristics of Function Graphs: **Function Theory: |x| and |x-2|**

Now the -1 is the only remaining thing to be taken care of. It means subtracting 1 from all points on |x-2|. In other words, it means pulling the graph vertically down by 1 unit. So, the new vertex would be (2,-1) instead of (2,0)

Characteristics of Function Graphs: **Function Theory: |x-2| – 1**

Which graph represents the functions f(X)= -|x-2| – 1 ?

Well, that should be quite easy after the analysis we just did. The only difference here is a minus sign before |x-2|. The minus sign simply inverts the graph of |x-2| with respect to the X axis. So, we can restart the previous problem just after the point where we had graph of |x-2|. But, this time before considering the -1, we shall invert the graph.

Characteristics of Function Graphs: **Graph of |x-2| and -|x-2|**

After this, we shall drag it down by one unit to incorporate the -1. And it is done.

The graph of a function must be linear if it has what characteristic?

What is a straight line? Normally it is defined as the minimum distance between two points on a plane surface. But it can also be defined from another angle. Since the X-Y plane is a collection of points, we can consider any line on this plane to be the locus or trace of a moving point, or a point whose X,Y co-ordinates are changing.

Moving along a straight line implies that the movement is happening without a change in direction. In other words, if a point starts moving from a given point and moves only in one given direction, then it is said to be following a straight line. So, if we are to express the linear graph as a function, then we must find an equation for the constant direction condition.

But how to express direction mathematically? Well, as we already have two axes of reference in the X-Y plane, a direction of a line may be expressed by the angle it makes with any of the two axes. So, let us assume that a straight line is inclined at an angle α. But that would mean a family of parallel lines and not just a single one. So, α cannot be the only parameter to a line.

Characteristics of Function Graphs: Family of lines with 45 degree slope

Note that the lines differ only in their Y intercept. The Y intercept is the distance from the origin of the point where the line meets the Y axis. Let us call this parameter, C. So, we have two parameters, α and C. Now, let us try to derive the equation of the line.

Characteristics of Function Graphs: **Intercept form of straight lines**

From the figure it should be clear from the right triangle, that for any point (x,y) on the line the governing condition has to be

(y-c)/x = tanα.

⟹ y = xtanα + c

⟹y = mx + c where m=tanα

Hence, any equation of the form y=ax+b must represent a straight line. In other words f(x) = ax + b is the desired form of a function in order to be linear.

The same can be derived also from the conventional definition of a straight line which states that a line is the shortest path between two points on a plane surface. So, let (x1,y1) and (x2,y2) be two points on a straight line.

Characteristics of Function Graphs: **Two points form of straight lines**

For any other point on the line, a condition can be derived by equating the slopes of the two line segments formed by the three points as the line must maintain it s slope at all segments. Hence the equation

(y-y₁)/(x-x₁)= (y₂-y₁)/(x₂-x₁)

⟹y(x₂-x₁) + x(y₁-y₂) + (x₁y₂-y₁x₂) = 0

This equation is of the form Ax + By + C = 0 which may be written in the form, y=ax+b, which we know as the form of a linear function.

Which graph is used to show change in a provided variable when a second variable is changed?

To draw an ideal graph of a function, we would need either a definite algebraic expression or infinite number of data points. In real life, both are not available most of the times. The data we have is scattered. In other words, we may have a list of (x,y) points which may be plotted on the graph, but the points may not be very densely located. But we have to connect those points anyways, as there is no other way to look at the pattern or the trend of the variables. A graph thus obtained is known as a line graph.

It is so named because neighboring points are joined with straight lines. This graph is best suited for illustrating a connection between two variables where one is depending on the other and are both changing. Time-series graphs are examples of line-graph where the X axis represents time in units of hours/days/months/years and the Y axis represents the variable whose value changes over time.

Sales	2010	2011	2012	2013	2014	2015	2016	2017	2018	2019
Year	4000	4700	4450	4920	5340	5120	5450	5680	5560	5900

Characteristics of Function Graphs

Periodic function

When the dependant variable repeats its value at a definite period or interval of the independent variable, the function is called periodic. The interval is called the period or fundamental period, sometimes as basic period or prime period also. The criteria for a function to be periodic is for some real constant T, f(x+T) = f(x). Which means f(x) is repeating its value after every T units of x. We may note the value of the function at any point, and we will find the same value at T units right and left to that point. That is the characteristic of a periodic function.

Characteristics of Function Graphs : **Sin(x) has a period of 2**

The above figure depicts the periodic behavior of Sinx. We take two random values of x, as x1 and x2 and draw lines parallel to the x axis from sin(x1) and sin(x2). We note that both the lines meet the graph again at a distance of exactly 2π. Hence the period of Sinx is 2π. So we can write sin(x+2 π) = sinx for any x. The other trigonometric functions are also periodic. Cosine has the same period as Sin and so do Cosec and Sec. Tan has a period π and so does Cot.

Which term gives the number of cycles of a periodic function that happen in one horizontal unit?

One full period is called a cycle. So, there is exactly one cycle in T units of x. Hence there are 1/T cycles in one unit of x. The number 1/T is of particular significance in the study of periodic functions since it tells how frequently the function is repeating its values. Hence the term ‘frequency’ is assigned to the number 1/T. Frequency is denoted by ‘f’, which is not to be confused with the ‘f’ of function The higher the frequency the more number of cycles are there per unit. Frequency and period are inversely proportional to each other, related as f = 1/T or T = 1/f. For Sin(X), the period is 2π, so frequency would be 1/2π.

Examples:

Calculate the period and frequency of Sin(3x)

As Sin(x) has one cycle in 2π, Sin(3x) will have 3 cycles in 2π as x progresses 3 times faster in Sin(3x). So frequency would be 3 times that of Sin(x) , that is 3/2π. That makes the period 1/(3/2π) = 2π/3

Calculate the period of Sin2x+sin3x

Note that any integer multiple of the fundamental period is also a period. In this problem, there are two components of the function. First has a period of π and the second one 2π/3. But these two are different, so neither can be the period of the composite function. But whatever is the period of the composition, it has to be a period of the components also. So, it has to be a common integer multiple to both of them. But there could be infinitely many of those. Hence the fundamental period would be the least common multiple of the periods of the components. In this problem that is Lcm(π,2π/3) = 2π

Characteristics of Function Graphs: **Period of a composite function**

Calculate the period of (Sin2x + Sin5x)/(Sin3x + Sin4x)

It is trivial but quite interesting to observe that the rule that we invented in the previous problem, does actually apply for any composition of periodic functions. So, in this case also the effective period would be the LCM of the periods of the components. That is LCM(π,2π/5,2π/3, π/2) = 2π

Calculate the period of Sinx + sin πx

At first, it seems obvious that the period should be LCM(2π,2), but then we realize that such a number does not exist as 2π is irrational so are its multiples and 2 is rational and so are its multiples. So, there could be no common integer multiple to these two numbers. Hence, this function is not periodic.

The fractional part function {x} is periodic.

f(x)={x}

This is known as the fractional part function. It leaves the greatest integer portion of a real number and leaves out only the fractional part. So, its value is always between 0 and 1 but never equal to 1. That graph should make it clear that it has a period 1.

Characteristics of Function Graphs : **The fractional part function {x}**

CONCLUSION

So far we discussed the Characteristics of Function Graphs. We should be now clear on the Characteristics and different types of graphs. We also had a idea of graphical interpretation of functions. Next article will be covering a lot more detail on concepts such as range and domain, inverse functions, various functions and their graphs, and a lot of worked out problems. To go deeper into the study, you are encouraged to read below

Calculus by Michael Spivak.

Algebra by Michael Artin.

For more mathematics article, please click here.

Function Theory: 9 Complete Quick Facts

December 31, 2023January 22, 2021 by TheMachineScience Core SME

INTRODUCTION

What is mathematics? Is it calculation? Is it logic? Is it symbols? Pictures? Graphs? Turns out, it is all of these and so much more. IT IS BUT A LANGUAGE. The universal language, having its symbols, characters, expressions, vocabulary, grammar, everything that makes a language, all perfectly reasoned, unique and unambiguous in their meaning. It is the language in which the laws of the universe are written. Hence it is the language we must learn and explore to unravel the mysteries of nature. We must begin our discussion on one of the most beautiful and fundamental mathematics topics, FUNCTION THEORY, with this philosophy.

WHAT ARE EXPRESSIONS, EQUATIONS AND, IDENTITIES?

Like all well-defined languages, mathematics comes with its own set of symbols and characters, numeric and alphabetical. An expression in mathematics is a combination of such symbols and characters. These all will be explain in this function theory discussion.

5+2/(9-3)

7a+2b-3c

2 cos 1/2 (α + β) cos 1/2 (α – β)

These are all mathematical expressions. No matter if they could be evaluated or not, if they are meaningful and if they follow proper syntax, they are expressions.

Now, when we compare two expressions with an ‘=’ sign, we have something like …

(1+x)² = 1+2x+x²

Which is an expression for equality of two expressions written on either side of an = sign. Note, that this equality is true for all values of x. These sorts of equalities are called IDENTITIES.

(1+x)² = 2+3x+2x²…………..(1)

Or like

(1+x)² = 7-3x+2x²…………(2)

Then they won’t be true for all values of x, rather they would be true for some values of x like (2) or they would be true for NO values of x, like (1). These are called EQUATIONS.

So to summarize, equalities that have for all values of the variables, are IDENTITIES. And equalities that hold for some or no values of the variables are EQUATIONS.

WHY DO WE NEED THE CONCEPT OF FUNCTION?

Is it not amazing that the universe is so perfectly balanced? A system of such enormous size made of so many smaller systems, each having so many variables interacting with each other, yet so well behaved. Does it not seem that everything is governed by a set of rules, unseen but existing everywhere? Take the example of the gravitational force. It is inversely proportional to the distance between bodies, and this rule is followed by all matters, everywhere in the universe. So, we must have a way to express such rules, such as connections between variables.

We are surrounded by such variables which depend on other variables. The length of the shadow of a building depends on its height and the time of the day. The distance travelled by car depends on the torque generated by its engine. It is the concept of function theory that enables us to express such relations mathematically.

SO WHAT IS A FUNCTION IN MATH?

Function Rule or FUNCTION as a rule

To put it simply, a function is a rule that binds two or more variables. If the variables are allowed to take only real values then it is simply an expression that defines a rule or a set of rules that assigns a real number to each of certain real numbers.

Now this definition surely requires some clarification which are given through the examples such as

1. The rule that assigns the cube of that number to each number.

f(x) = x³

2. The rule that assigns (x²-x-1)/x³ to each x

f(x) = (x²-x-1)/x³

3. The rule that assigns (x²–x-1)/(x²+x+1) to all x which are not equal to 1 and the number 0 to 1

f(x) = (x2-x-1)/(x2+x+1) for x ≠ 1

= 0 for x=1

f(x) = x² for -1 < x < π/3

The rule that assigns

2 to number 5

3 to number 8/3

π/2 to number 1

and to the rest

The rule that assigns to a number x, the number of 1s in its decimal expansion if the count is finite and 0 if there are infinitely many 1s in the expansion.

These examples should make one thing very clear that a function is any rule that assigns numbers to specific other numbers. These rules may not always be expressible by algebraic formulation. These may not even point to one unique condition that applies to all numbers. And it doesn’t have to be a rule that one can find in practice or in the real world, like the one in rule 6. No one can tell which number this rule assigns to the number π or √2. The rule also may not apply to some numbers. For example, rule 2 does not apply to x=0. The set of numbers to which the rule applies is called the DOMAIN of the function.

SO WHAT DOES y= f(x) MEAN?

Note, that we are using the expression y=f(x) to write a function. Whenever we start an expression with ‘f(x) = y’ then we mean that we are about to define a function that relates a set of numbers with a set of values of the variable x.

FUNCTION as a relation

So, in other words, and perhaps in a more general sense, a function is a relation between two sets A and B, where all the elements in the set A have an element assigned to them from the set B. The elements from set B are called the IMAGES and the elements of set A are called the PRE-IMAGES.

The process of relating the elements is called MAPPING. Of course there could be many ways in which these mappings can be done, but we would not call all of them as functions. Only those mappings that relate the elements in such a way that every element in set A has exactly one image in set B, are to be called functions. It is sometimes written as f : A–> B . This is to be read as ‘f is a function from A to B’.

The set A is called the DOMAIN of the function and the set B is called the CO-DOMAIN of the function. If f is such that the image of one element a of set A is the element b from set B, then we write f(a) = b, read as ‘f of a is equal to b’, or ‘b is the value of f at a’, or ‘b is the image of a under f’.

TYPES OF FUNCTIONS

Functions may be classified as per the way they relate the two sets.

One – one or injective function

Image1 Types of Functions — function theory: One to One or injective function

The figure says it all. It is when a function relates every element of a set to a unique element of another set, it is a one to one or injective function.

Many – one function

Again, the figure is quite self-explanatory. Evidently there are more than one pre-image to a particular image. Hence the mapping is many to one. Note, that it does not violate the definition of a function as no element from set A has more than one image in set B.

ONTO function or SURJECTIVE function

Image3 Onto functions 1 — Function Theory: ONTO function or SURJECTIVE function

When all the elements of set B has at least one pre-image, then the function is called Onto or surjective. Onto mapping can be one to one or many to one. The one depicted above is evidently many to one onto mapping. Note that the picture used previously for depicting one to one mapping is also onto mapping. This sort of one to one onto mapping is also known as BIJECTIVE mapping.

Into function

Image4 onto function2 — Function Theory: INTO Function

When there is at least one image without any pre-image, it is an INTO function. Into function can be one to one or many to one. The one depicted above is obviously one to one into.

GRAPH OF A FUNCTION

As it is said earlier that a function assigns real numbers to certain real numbers, it is quite possible and convenient to plot the pair of numbers on X-Y Cartesian plane. The trace obtained by connecting the points, is the graph of the function.

Let us consider a function f(x) = x + 3. Then, we could evaluate f(x) at x=1,2,3 to obtain three pairs of x and f(x) as (1,4) , (3,6) and (5,8). Plotting these points and connecting them shows that the function traces a straight line in the x-y plane. This line is the graph of the function.

Image5 graph of a function1 — Function Theory: Graph of a function_1

Evidently, the nature of the trace will vary according to the expression for the function. Thus we get a range of graphs for different kind of expressions. A few are given.

The graphs of f(x) = sin x, f(x) = x²and f(x) = e^x from left to right

Image6 graphof function2 — Function Theory: Graph of a function_2

At this point, one can see that the expression for a function actually looks like that of an equation. And it is true, for example y = x + 3 is indeed an equation as well as a function definition. This brings us the question, are all equation functions? If not then

How to tell if an equation is a function?

All the equations depicted in the graphs earlier are actually functions, as for all of those, there is exactly one value of f(x) or y for some value of x. This means that the expression for f(x) should yield only one value when evaluated for any value of x. This is true for any linear equation. But if we consider the equation y²= 1-x², we find that there are always two solutions for all x within 0 to 1, in other words, two images are assigned to each value of x within its range. This violates the definition of a function and hence cannot be called a function.

This should look clearer from the graph that there are exactly two images of each x as a vertical line drawn at any point on the x axis will cut the graph at exactly two points.

Image7 graph of function3 — Function Theory: Graph of a function_3

So, this brings us to one important conclusion that not all equations are functions. And whether an equation is a function, can be verified by the vertical line test, which is simply imagining a variable vertical line at each point on x axis and seeing if it meets the graph at a single point.

This also answers another important question, which is, how to tell if a function is one to one? Surely enough, that answer is also in the graph and can be verified by the vertical line test.

Now, one could ask if there is a way to tell the same without obtaining the graph or if it could be told algebraically as it is not always easy to draw graphs of functions. Well the answer is yes, it can be done simply by testing f(a)=f(b) implies a=b. This is to say that even if f(x) takes the same value for two values of x, then the two values of x cannot be different. Let us take an example of the function

y=(x-1)/(x-2)

As one would notice that it is difficult to plot the graph of this function as it is non-linear in nature and does not fit the description of any familiar curve and moreover is not defined at x=2 . So, this problem definitely calls for a different approach from the vertical line test.

So, we begin by letting

f(a)=f(b)

=> (a-1)/(a-2)=(b-1)/(b-2)

=>(a-1)(b-2)=(b-1)(a-2)

=>ab-2a-b+2=ab-2b-a+2

=> 2a+b=2b+a

=>2(a-b)=(a-b)

This is only possible for a-b=0 or a=b

So, the function is indeed one to one, and we have proved it without graphing.

Now, we would want to see when some function fails this test. We might want to test equation of the circle we tested before. We start by writing

f(a)=f(b)

f(x) = x²

=> a²=b²

a² =b²

=> a=b or a=-b

Which simply means that there are solutions other than a=b, hence f(x) is not a function.

IS IT SO DIFFICULT TO PLOT y=(x-1)/(x-2) ?

We are going to discuss graphing of a function in much greater detail in the upcoming articles but here it is necessary to get familiar with the basics of graphing as it helps immensely with problem solving. A visual interpretation of a calculus problem often makes the problem very easy and knowing how to graph a function is the key to a good visual interpretation.

So, to plot the graph of (x-1)/(x-2), we begin by making a few critical observations such as

1. The function becomes 0 at x=1.

2. The function becomes undefined at x=2 .

3. The function is positive everywhere except for 1<x<2.

Because in this interval (x-1) is positive and (x-2) is negative, this makes their ratio negative.

4. As x goes to -∞ the function nears unity from the lower side, meaning that it goes close to 1 but is always less than 1.

Because for x<0, (x-1)/(x-2) =(|x|+1)/(|x|+2)<1 as |x|+2>|x|+1

5. As x goes to +∞ the function nears unity from the upper side, meaning that it goes close to 1 but is always greater than 1.

6. As x goes to 2 from the left side, the function goes to -∞.

7. As x goes to 2 from the right side, the function goes to +∞.

8. The function is always decreasing for x>2.

PROOF:

We take two close values of x as (a, b) such that (a, b) >2 and b>a

now, f(b) – f(a)

=(b-1)/(b-2)-(a-1)/(a-2)

={(b-1)(a-2)-(a-1)(b-2)}/(a-2)(b-2)

=(a-b)/{(a-2)(b-2)}

<0 as (a-b)<0 for b>a

and (a-2)(b-2)> 0 as (a, b)> 2

This implies f(b)<f(a) for all a>2, in other words f(x) is strictly decreasing for x>2

9. The function is always decreasing for x<2
PROOF: same as before. We leave it for you to try.

Combining these observations makes the graphing quite easy. Combining 4,9 and 6 we can say that as x goes from -∞ to 2, the trace starts from unity and falls gradually to touch 0 at x=1 and falls further to -∞ at x=2. Again combining 7,5 and 8 it is easy to see that as x goes from 2 to +∞, the trace starts falling from +∞ and keeps getting close to unity never really touching it.

This makes the complete graph look like

Image8 graph of Function4 1 — Function Theory: Graph of a function_4

Now it becomes evident that the function is indeed one to one.

CONCLUSION

So far we discussed the basics of function theory. We should be now clear on the definitions and types of functions. We also had a little idea of graphical interpretation of functions. Next article will be covering a lot more detail on concepts such as range and domain, inverse functions, various functions and their graphs, and a lot of worked out problems. To go deeper into the study, you are encouraged to read

Calculus by Michael Spivak.

Algebra by Michael Artin.

For more mathematics article, please click here.

Coordinate Geometry: 3 Things Most Beginner’s Don’t Know

December 31, 2023December 31, 2020 by NASRINA PARVIN

Coordinate Geometry

Today we are here to discuss Coordinate Geometry from the root of it. So, The whole article is about what Coordinate Geometry is, relevant problems and their solutions as much as possible.

(A) Introduction

Coordinate Geometry is the most interesting and important field of Mathematics. It is used in physics, engineering and also in aviation, rocketry, space science, spaceflight etc.

To know about Coordinate Geometry first we have to know what Geometry is.
In greek ‘Geo’ means Earth and ‘Metron’ means Measurement i.e. Earth Measurement. It is the most ancient part of mathematics, concerned with the properties of space and figures i.e positions, sizes, shapes, angles and dimensions of things.

What is Coordinate Geometry?

Coordinate geometry is the way of learning of geometry using the Co-ordinate system. It describes the relationship between geometry and algebra.
Many mathematicians also called Coordinate geometry as Analytical Geometry or Cartesian Geometry.

Why is it called Analytical Geometry?

Geometry and Algebra are two different branches in Mathematics. Geometrical shapes can be analyzed by using algebraic symbolism and methods and vice versa i.e. algebraic equations can be represented by Geometric graphs. That is why it is also called Analytical Geometry.

Why is it called Cartesian Geometry?

Coordinate Geometry was also named Cartesian Geometry after French mathematician Rene Descartes as he independently invented the cartesian coordinate in the 17th century and using this, put Algebra and Geometry together. For such a great work Rene Descartes is known as the Father of Coordinate Geometry.

(B) Coordinate system

A Coordinate system is the base of Analytical Geometry. It is used in both two dimensional and three-dimensional fields. There are four types of coordinate system in general.

(C) The whole subject of coordinate Geometry is divided into two chapters.

One is ‘Coordinate Geometry in Two Dimensions’.
The second one is ‘Coordinate Geometry in Three Dimensions’.

Coordinate Geometry in Two Dimensions (2D):

Here we are going to discuss both the Cartesian and Polar Coordinates in two dimensions one by one. We will also solve some problems to get a clear idea of the same, and later we will find the relation between them as well.

Cartesian Coordinate in 2D:

At first, we will have to learn the following terms through graphs.
i) Coordinate Axes
ii) Origin
iii)Coordinate Plane
iv) Coordinates
v) Quadrant

Read and Follow the Figures simultaneously.

Suppose the horizontal line XXand vertical line YY are two perpendicular lines intersecting each other at right angles at the point O , XXand YY are number lines, the intersection of XXand YY forms XY-plane and P is any point on this XY-plane.

Coordinate Axes in 2D

Here XX and YY are described as the Coordinate Axes. XX is indicated by X-Axis and YY is indicated by Y-Axis. Since XX and YY are number lines, the distances measured along OX and OY are taken as positive and also the distances measured along OX and OY are taken as negative. (See above graph.1)

What is Origin in 2D?

The point O is called the Origin. O is always supposed to be the starting point. To find the position of any point on the coordinate plane we always have to begin the journey from the origin. So the origin is called the Zero Point. (Please refer the above graph.1)

What do we understand by a Coordinate Plane?

The XY plane defined by two number lines XX and YY or the X-axis and Y- axis is called the Coordinate Plane or Cartesian Plane. This Plane extends infinitely in all direction. This is also known as two-dimensional plane. (See above graph.1)

*Assume the variables x>0 and y>0 in the above figure.

What is Coordinate in 2D?

Coordinate is a pair of numbers or letters by which the position of a point on the coordinate plane is located. Here P is any point on the coordinate plane XY. The coordinates of the point P is symbolized by P(x,y) where x is the distance of P from Y axis along X axis and y is the perpendicular distance of P from X axis respectively. Here x is called the abscissa or x-coordinate and y is called the ordinate or y-coordinate (See above Graph 2)

How to Plot a Point on the coordinate plane?

Always we will have to start from the origin and first walk towards right or left along X axis to cover the distance of x-coordinate or abscissa ,then turn the direction up or down perpendicularly to the X axis to cover the distance of ordinate using units and their signs accordingly. Then we reach the required point .

Here to represent the given point P(x,y) graphically or to plot it on the given XY plane, first start from the origin O and cover the distance x units along X axis (along OX) and then turn at 90 degree angle with X axis or parallelly to Y axis(here OY) and cover the distance y units . (See above graph 3)

How to find coordinates of a given point in 2D ?

Let XY be the given plane,O be the origin and P be the given point.
First draw a perpendicular from the point P on X axis at the point A. Suppose OA=x units and AP=y units, then the Coordinates of the point P becomes (OA , AP) i.e. (x,y).

Similarly if we draw another perpendicular from the point P on Y axis at the point B, then BP=x and OB=y.
Now since A is the point on the X axis ,the distance of A from Y axis along X axis is OA=x and perpendicular distance from X axis is zero,so the coordinates of A becomes (x,0).
Similarly, the coordinates of the point B on the Y axis as (0,y) and the coordinates of Origin O is (0,0).

Graph 5 * colour green denotes the beginning

What is Quadrant in 2D?

Coordinate Plane is divided into four equal sections by the coordinate axes. Each section is called Quadrant. Going around counterclockwise or anticlockwise from upper right, the sections are named in the order as Quadrant I, Quadrant II, Quadrant III and Quadrant iv.

Here we can see the X and Y axes divide the XY plane into four sections XOY, YOX, XOY and YOX accordingly. Therefore, the area XOY is the Quadrant I or first quadrant, YOX is the Quadrant II or second quadrant, XOY is the Quadrant III or third quadrant and YOX is the Quadrant IV or fourth quadrant.(please refer the graph 5)

Points in Different Quadrants of coordinate plane:

Since OX is +ve and OX is -ve side of X axis and OY is +ve and OY is -ve side of Y axis, signs of coordinates of points in different quadrants—-
Quadrant I: (+,+)
Quadrant II: (-,+)
Quadrant III: (-,-)
Quadrant IV: (+,-)

For example, if we go along OX from O and draw a perpendicular from any point P in the Quadrant I on the X axis (OX) at the point A so that OA=x and AP=y then coordinate of P is defined as (x,y) as described in the article (How to find coordinate of a given point?).

Again if we go along OX from O and draw a perpendicular from any point Q in the Quadrant II on the X axis (on OX) at the point C so that OC=x and CQ=y then the coordinates of Q is defined as (-x,y).
Similarly the coordinates of any point R in quadrant III is defined as (-x,-y) and the coordinates of any point in quadrant IV is defined as (x,-y). (see graph 6)

Conclusion

The brief information about Coordinate Geometry with basic concepts has been provided to get a clear idea to start the subject. We will subsequently discuss details about 2D and 3D in the upcoming posts. If you want further study go through:

Reference

1. https://en.wikipedia.org/wiki/Analytic_geometry
2. https://en.wikipedia.org/wiki/Geometry

For more topics on Mathematics, please follow this Link .

15 Examples Of Permutations And Combinations

December 31, 2023December 31, 2020 by DR. MOHAMMED MAZHAR UL HAQUE

Illustration of the concept Permutations and Combinations by the examples

In this article, we have discussed some examples which will make the foundation strong of the students on Permutations and Combinations to get the insight clearance of the concept, it is well aware that the Permutations and combinations both are the process to calculate the possibilities, the difference between them is whether order matters or not, so here by going through the number of examples we will get clear the confusion where to use which one.

The methods of arranging or selecting a small or equal number of people or items at a time from a group of people or items provided with due consideration to be arranged in order of planning or selection are called permutations.

Each different group or selection that can be created by taking some or all of the items, no matter how they are organized, is called a combination.

Basic Permutation (nPr formula) Examples

Here We are making group of n different objects, selected r at a time equivalent to filling r places from n things.

The number of ways of arranging = The number of ways of filling r places.

ⁿP_r= n. (n-1). (n-2)…(n-r+1) = n/(n-r)!

so nPr formula we have to use is

ⁿP_r= n!/(n-r)!

Example 1): There is a train whose 7 seats are kept empty, then how many ways can three passengers sit.

solution: Here n=7, r=3

so Required number of ways=

ⁿP_r= n!/(n-r)!

⁷P₃ = 7!/(7-3)! = 4!.5.6.7/4! = 210

In 210 ways 3 passengers can sit.

Example 2) How many ways can 4 people out of 10 women be chosen as team leaders?

solution: Here n=10, r=4

so Required number of ways=

ⁿP_r= n!/(n-r)!

¹⁰P₄ = 10!/(10-4)! = 6!7.8.9.10/6! = 5040

In 5040 ways 4 women can be chosen as team leaders.

Example 3) How many permutations are possible from 4 different letter, selected from the twenty-six letters of the alphabet?

solution: Here n=26, r=4

so Required number of ways=

ⁿP_r= n!/(n-r)!

²⁶P₄ = 26!/(26-4)! = 22!.23.24.25.26/22! = 358800

In 358800 ways, 4 different letter permutations are available.

Example 4) How many different three-digit permutations are available, selected from ten digits from 0 to 9 combined?(including 0 and 9).

solution: Here n=10, r=3

so Required number of ways=

ⁿP_r= n!/(n-r)!

¹⁰P₃ = 10!/(10-3)! = 7!.8.9.10/7! = 720

In 720 ways, three-digit permutations are available.

Example 5) Find out the number of ways a judge can award a first, second, and third place in a contest with 18 competitors.

solution: Here n=18, r=3

so Required number of ways=

ⁿP_r= n!/(n-r)!

¹⁸P₃ = 18!/(18-3)! = 15!.16.17.18/15! = 4896

Among the 18 contestants, in 4896 number of ways, a judge can award a 1st, 2nd and 3rd place in a contest.

Example

6) Find the number of ways, 7 people can organize themselves in a row.

solution: Here n=7, r=7

so Required number of ways=

ⁿP_r= n!/(n-r)!

⁷P₇= 7!/(7-7)! = 7!/0! = 5040

In 5040 number of ways, 7 people can organize themselves in a row.

Examples based on Combination (nCr formula/ n choose k formula)

The number of combinations (selections or groups) that can be set up from n different objects taken r (0<=r<=n) at a time is

This is commonly known as nCr or n choose k formula.

ⁿC_k = n!/k!(n-k)!

Examples:

1) If you have three dress with different colour in red, yellow and white then can you find a different combination you get if you have to choose any two of them?

Solution: here n=3, r=2 this is 3 CHOOSE 2 problem

ⁿC_r = n!/r!(n-r)!

³C₂ = 3!/2!(3-2)! = 2!.3/2!.1 = 3

In 3 different combination you get any two of them.

2) How many different combinations can be done if you have 4 different items and you have to choose 2?

Solution: here n=4, r=2 this is 4 CHOOSE 2 problem

ⁿC_r = n!/r!(n-r)!

⁴C₂ = 4!/2!(4-2)! = 2!.3.4/2!.2! = 6

In 6 different combination you get any two of them.

3) How many different combinations can be made if you have only 5 characters and you have to choose any 2 among them?

Solution: here n=5, r=2 this is 5 CHOOSE 2 problem

ⁿC_r = n!/r!(n-r)!

⁵C₂ = 5!/2!(5-2)! = 3!.4.5/2!.3! = 10

In 10 different combination you get any two of them.

4) Find the number of combinations 6 choose 2.

Solution: here n=6, r=2 this is 6 CHOOSE 2 problem

ⁿC_r = n!/r!(n-r)!

⁶C₂ = 6!/2!(6-2)! = 4!.5.6/2!.4! = 15

In 15 different combination you get any two of them.

5) Find the number of ways of choosing 3 members from 5 different partners.

Solution: here n=5, r=3 this is 5 CHOOSE 3 problem

ⁿC_r = n!/r!(n-r)!

⁵C₃ = 5!/3!(5-3)! = 3!.4.5/3!.2! = 10

In 10 different combination you get any three of them.

6) Box of crayons having red, blue, yellow, orange, green and purple. How many unlike ways can you use to draw only three colour?

Solution: here n=6, r=3 this is 6 CHOOSE 3 problem

ⁿC_r = n!/r!(n-r)!

⁶C₃ = 6!/3!(6-3)! = 3!.4.5.6/3!.3.2.1 =20

In 20 different combination you get any three of them.

7) Find the number of combinations for 4 choose 3.

Solution: here n=4, r=3 this is 4 CHOOSE 3 problem

ⁿC_r = n!/r!(n-r)!

⁴C₃ = 4!/3!(4-3)! = 3!.4/ 3!.1! = 4

In 4 different combination you get any three of them.

8) How many different five-person committees can be elected from 10 people?

Solution: here n=10, r=5 this is 10 CHOOSE 5 problems

ⁿC_r = n!/r!(n-r)!

¹⁰C₅ = 10!/5!(10-5)! = 10!/5!.5! = 5!.6.7.8.9.10/5!.5.4.3.2 = 7.4.9 =252

So 252 different 5 person committees can be elected from 10 people.

9) There are 12 volleyball players in total in college, which will be made up of a team of 9 players. If the captain remains consistent, the team can be formed in how many ways.

Solution: here as the captain already has been selected, so now among 11 players 8 are to be chosen n=11, r=8 this is 11 CHOOSE 8 problem

ⁿC_r = n!/r!(n-r)!

¹¹C₈ = 11!/8!(11-8)! = 11!/8!.3! = 8!.9.10.11/8!.3.2.1 = 3.5.11 = 165

So If the captain remains consistent, the team can be formed in 165 ways.

10) Find the number of combinations 10 choose 2.

Solution: here n=10, r=2 this is 10 CHOOSE 2 problem

ⁿC_r = n!/r!(n-r)!

¹⁰C₂ = 10!/2!(10-2)! = 10!/2!.8! = 8!.9.10/2!.8! = 5.9 = 45

In 45 different combination you get any two of them.

We have to see the difference that nCr is the number of ways things can be selected in ways r and nPr is the number of ways things can be sorted by means of r. We have to keep in mind that for any case of permutation scenario, the way things are arranged is very very important. However, in Combination, the order means nothing.

Conclusion

A detailed description with examples of the Permutations and combinations has been provided in this article with few real-life examples, in a series of articles we will discuss in detail the various outcomes and formulas with relevant examples if you are interested in further study go through this link.

Reference

SCHAUM’S OUTLINE OF Theory and Problems of DISCRETE MATHEMATICS
https://en.wikipedia.org/wiki/Permutation
https://en.wikipedia.org/wiki/Combination

Permutations and Combinations: 11 Facts You Should Know

January 1, 2024December 29, 2020 by DR. MOHAMMED MAZHAR UL HAQUE

Permutations and Combinations

Permutations and Combinations, this article will discuss the concept of determining, in addition to the direct calculation, the number of possible outcomes of a particular event or the number of set items, permutations and combinations that are the primary method of calculation in combinatorial analysis.

Common mistakes while learning Permutations and Combinations

There is always confusion amongst the student between permutations and combinations because both are related to the number of the arrangement of different objects and the number of the possible outcome of a particular event or number of ways to get an element from a set. The topic of permutation & combination with examples and the difference between them with justification will be discussed here.

A simple and handy technique to remember the difference between the permutations and combinations is: a permutation is related with the order means the position is important in permutation while the combination is not related with the order means the position is not important in combination.

Before the discussion of permutations and combinations, we require some prerequisites, which are frequently used.

What is Factorial

Factorial is the product of the positive integers from 1 to n (counting 1 and n) denoted by n! and read as n factorial is described as below

n! = 1.2.3.4… (n-2).(n-1).n = n.(n-1).(n-2)…3.2.1

ⁿP_r = n.(n-1).(n-2)…(n–r+1) = n!/(n-r)!

Mind it 0!=1

0! = 1

1! = 1

n! = n(n-l)!

e.g 3! = 3.2.1 = 6

4! = 4.3.2.1 = 24

5! = 5.4! = 5.24 = 120

Counting Methods (Principle of Multiplication and addition)

Principle of addition: If no two events can happen at the same time, then one of the events can happen in

n1 + n2 + n3 +・ ・ ・.ways

Principle of Multiplication: Considering that if the events occurred one after the other, then all the events can happen in the order indicated in:

n₁.n₂.n₃…ways

Example: If an Institute runs 7 different art courses, 3 different technical courses, and 4 different physical courses.

If a student wants to enroll one of each type of course then the number of ways would be

m=7.3.4=84

If a student wants to enroll just one of the courses, then the number of ways would be

n=7 + 3 + 4=14

What is Permutation

The different positioning of the objects are called Permutations, where the order of the arrangement matters. Any positioning of a set of n different objects in a given order is called a permutation of the object.

Consider an example of the set of letters {P,Q,R,S}, then

Some of the permutations of the four alphabets taken 4 at a glance are QSRP, SRQP and PRSQ

Any ordering of any r<=n of these particular objects in a specific order is called an “r-permutation” or “a permutation of the n objects taken r at a time.

Basically we like those number of such permutations without set down them.

Example of Permutation Formula

The number of permutations of n different objects taken r at a time will be indicated by

ⁿP_r= n. (n-1).(n-2)…(n-r+1) = n!/(n–r)!

In mathematics this is denoted by different ways, some of them are mentioned below:

P(n,r), nPr,Pn,r ,or (n)r

EXAMPLE: Calculate the number m of permutations of six objects, say A, B, C, D, E, F taken three at a glance.

Solution: Here n=6, r=3, m=?

ⁿP_r = n!/(n-r)!

m = ⁶P₃ = 6!/(6-3)! = 6!/3! = 3!.4.5.6/3!= 4.5.6 = 120

So m=120

EXAMPLE: How many words can be generated by using 2 letters from the word “MATHS”?

Solution: Here n=5, r=2, m=?

ⁿP_r = n!/(n-r)!

m = ⁵P₂= 5!/(5-2)! = 5!/3! = 3!.4.5/3! = 4.5 = 20

so the required number of words are 20.

What do you understand by a Combination?

A combination for n different elements taken r at a time is any selection of r-th elements where orders are not being considered. Such a selection is called an r-combination. In brief, a Combination is a selection in which the order of the objects selected is not important.

The Combination gives the number of ways a particular set can be arranged, where the order of the arrangement does not matter.

To understand the situation of Combination, consider the example

Twenty people arrive in a hall and everyone shakes hand with all the others. How can we get the number of handshakes? “A” shaking hands with B and B with A will not be two different handshakes. Here, the order of handshake is not important. The number of handshakes will be the combinations of 20 different things taken 2 at a time.

Combination Formula with a simple example

The number of such combinations will be denoted by

Sometimes it is also denoted by C(n,r), ⁿC_r , C_n,r or C_rⁿ

Example: A class contains 10 students with 6 men and 4 women. Find the number n of ways to choose a 4-member committee among those students.

This is related to combinations, not permutations, since order is not an important factor in a committee. There are “10 choose 4” such committees. That is:

here n=10, r= 4

so in 210 ways we can choose such 4-member committee.

Example: A container has 6 blue balls and 8 red balls. Identify the number of ways two balls of any of the colors can be drawn from the container.

Here possibly “14 choose 2” ways for selecting 2 of the 14 balls. Thus:

Here n=14 , r=2

gif.latex?%5E%7B14%7DC %7B2%7D%20%3D%20%5Cbinom%7B14%7D%7B2%7D%20%3D%20%5Cfrac%7B14%21%7D%7B2%21%2814 2%29%21%7D%20%3D%20%5Cfrac%7B14.13.12.%21%7D%7B2.1

so in 91 ways two balls can be drawn of any color.

Difference between Permutation and Combination

The difference between permutation vs combination is briefly given here

Permutation	Combination
Order is Important	Order is not Important
Order counts	Order does not count
Used for arrangements like electing president, vice president, and treasurer	Used for selection like selecting teams and committee without positions
For electing first, second and third specific positions	For selecting any three random
For arranging the cards or balls with position and color	For selecting any color and position

Difference between Permutations and Combinations

Where to apply Permutations and Combinations

This is the important step that should be kept in mind that whenever the situation is for arrangement, ordering and uniqueness we have to use Permutation and whenever the situation is for selection, choosing, picking and combination without the concern of order we have to use Combination. If you keep these basics in your mind there will be no confusion “what to use and what not” whenever a question arises.

Use of Permutations and Combinations in real life with examples

In real life permutation and the combination is used in almost everywhere because we know that in real life there would be a situation when order is important and somewhere order is not important, in those situations we have to use the corresponding method.

For example

Find the number N of teams of 11 with a given captain that can be selected from 26 players.

Frequently Asked Questions – FAQs

What is factorial?

The product of the positive integers from 1 to n (including 1 & n )

n! = 1.2.3… (n-2). (n-1). n

What is a permutation?

The different ordering of the objects are called Permutations

What is a Combination?

The Combination provides the number of ways a specific set can be set out, where the order of the arrangement does not matter.

Application of permutations and combinations in practical life

A Permutation is used for arrangement or selection of lists where the order is important, and Combination is used for selection or choice where the order is not important.

Permutation formula

ⁿP_r = n!/(n-r)!

Combination formula

Is there any relation between permutations and Combinations?

Yes,

ⁿC_r= ⁿP_r/r!

Can we use Permutations and combinations in real life?

Yes,

In the arrangement of words, alphabets, numbers, positions and colours etc. where the order is important permutation will be used

In the selection of committee, teams, menu, and subjects etc where the order is not important combination will be used.

The brief information about permutations and combinations with basic formula is given read twice or thrice till you get the idea about the concept, in consecutive articles we will discuss in detail the different results and formulae with suitable examples of permutations and combinations. If you want further study go through:

For more Topics on Mathematics, please follow this link.

References:

1. SCHAUM’S OUTLINE OF Theory and Problems of DISCRETE MATHEMATICS

2. https://en.wikipedia.org/wiki/Permutation

3. https://en.wikipedia.org/wiki/Combination

4. https://in.bgu.ac.il/

5. https://www.cs.bgu.ac.il/