Permutation and Combination﻿- Formula, Difference, Examples, Theorem

SOURCE: Don’t Memorise

Permutation and Combination is a mathematical formula used in various ways in which a set of objects are selected and are formed into a subset.

This arranging of subsets is known as permutation when the order is the main factor and combination when the order is not a factor.

Permutation and Combination Formula

The formula of permutation and combination has ‘r’ as an element out of ‘n’ number of elements which is total elements. ‘P’ and ‘C’ are permutation and combination of the objects.

Permutation and combination deal with the counting and arrangement of a particular set of data. Factorial (n!): It is the product of all positive integers less than or equal to n. Example: 4! = 4 × 3 × 2 × 1 = 24.

Note: 0! = 1

Permutation refers to the act of arranging all the elements of a set into a sequence or order. It is denoted by nPr. A permutation is the choice of r things from a set of n things without replacement and where the order matters.

nPr = (n!) / (n-r)!

Example: Arrange the given 3 numbers 1, 2, 3, taking two at a time. The numbers can be arranged in 6 ways: (12, 21, 13, 31, 23, 32).Note: 12 and 21, 13 and 31 or 23 and 32 do not mean the same, because the order of numbers is important.

Formulas for Permutation and combination is as follows:

Permutation Formula

Permutation arranges all the objects accordingly while subsets out of it. There is one easy and tricky way to get the number of letters for which the permutation is required and it is n P n = n! The required formula is mentioned below n P r = n! / (n – r)!. To understand more about
Permutation, go through Brett Berry’s post.

Non-Circular Permutation

Arrangement of ‘n‘ distinct objects:

Example: How many different ways can the letters of the word PRISM be arranged?
Solution: There are 5 distinct letters. The total number of arrangements is 5! = 120

Arrangement of ‘n‘ objects in which some elements are repeated:

Example: How many different ways can the letters of the word INDIA be arranged?
Solution: There are 5 letters, 2 among them are repeated. To avoid duplication in the counting, divide the total number of arrangements by 2!. The total number of arrangements is 5! / 2! = 120/2 = 60

Arrangement of ‘n‘ objects on a conditional basis:

Example: How many different ways can the letters from the word LONG be re-arranged so that the word starts with a vowel?
Solution: Out of the 4 distinct letters, only 1 letter (O) is a vowel. The first letter can be arranged in only one way and the remaining 3 places can be arranged in 3! ways. The total number of arrangements is 1 x 3! = 6

Cyclic Permutation

Permutation around a circular object is done by fixing one object and permuting the remaining objects.

Example: How many different ways can 6 students be seated around a circular table?
Solution: The number of circular permutations of different items taken all at a time is (n – 1)! = (6 – 1)! = 120

Combination Formula

The combination is different from Permutation and it’s formula remains as it is i.e. n C r = n! / [(r !)(n – r)!]A combination is the choice of things from a set of n things without replacement and where order does not matter.

nCr = n!/ r! × (n-r)!

Example: If we have to select two balls out of 3 balls, X, Y, Z, then find the number of combinations possible.
Solution:Only two balls are to be selected and arranged. Therefore, this is possible in 3 different ways: (XY, YZ, XZ,).

Note: XY and YX are the same combinations.

Difference between Permutation and Combination

 Permutation Combination The order of elements is taken into consideration In combination, the order does not matter n P r = n! / (n – r)! n C r = n! / [(r !)(n – r)!] There are different ways in which a collection of elements can be arranged Whereas in combination we cannot

Permutation and Combination Questions

There are various questions solved in this particular chapter and a student needs to learn the trickest way to solve it for various entrances and competitive exams.

There are some various questions for the students with the solution:

Q. There are 7 consonants and 4 vowels. How many words of 3 consonants and 2 vowels can be formed?

Ans. Number of ways of selecting 3 consonants from 7= 7 C3

Number of ways of selecting 2 vowels from 4= 4C2

The total number of ways is 7 C3 x 4C2 = 210

Number of ways of arranging 5 letters is 5! =120

Required number of ways is 210 x 120 = 25200

Q. In how many of the ways can a group of 5 men and 2 women be made out of 7 men and 3 women?

Ans. Number of ways of selecting 5 men from 7 = 7C5

Number of ways of selecting 2 women from 3 = 3C2

Required number of ways is by multiplying both the solutions using
nCr = nC(n-r) = 63

Q. How many 3 letter words with and without meaning can be formed out of the letters of the word BACKGROUND if repetition of letters is not allowed?

Ans. The word has 10 different letters

The number of 3 letter words with and without meaning formed by using these letters = 10P3 = 10 x 9 x 8 = 720

Theorem of Counting

1. Rule of Addition: If the first task is performed in x ways and the second task is performed in y ways, then either of the two can be performed in (x + y) ways.
2. Rule of Multiplication: If the first task is performed in x ways and the second task is performed in y ways, then both of the two can be performed in (x × y) ways.

Check Out:

FAQ

What is the difference between combination and permutation?

Ans. 1. In permutation, the order of elements is taken into consideration while in combination, the order does not matter.
2. In permutation, there are different ways in which a collection of elements can be arranged whereas in combination we cannot.

Example: If your password is 1234 and if you enter 4321 as your password, it won’t open because the order is different i.e permutation.

When to use permutation and when to use combination?

Ans. Permutation and combination deal with the counting and arrangement of a particular set of data. Factorial (n!): It is the product of all positive integers less than or equal to n. Example: 4! = 4 × 3 × 2 × 1 = 24.
Permutationrefers to the act of arranging all the elements of a set into a sequence or order. It is denoted by nPr. A permutation is the choice of r things from a set of n things without replacement and where the order matters.

How do you solve permutations and combinations easily?

Ans. This arranging of subsets is known as permutation when the order is the main factor and combination when the order is not a factor.
See this video so that you can solve permutations and combinations easily: Quick Ways of Doing Permutations and Combinations

How many unique combinations are there?

Ans. According to the fundamental counting principle if you want to find the unique combinations of 2 independent events just multiply the number of ways each event can happen together.
Example: if you have 5 shirts and 7 pants then the unique combination will be 5 * 7 = 35 unique combination.

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Rachit Agrawal

Rachit believes in the power of education and has studied from the top institutes of IIIT Allahabad, IIM Calcutta, and Francois Rabelias in France. He has worked as Software Developer with Microsoft and Adobe. Post his MBA, he worked with the world’s # 1 consulting firm, The Boston Consulting Group across multiple geographies US, South-East Asia and Europe.