Permutation and Combination

Permutation and combination are the ways to select certian objects from a group of objects to form subsets with or without replacement. It defines the various ways to arrange a certain group of data. When we select the data or objects from a certain group, it is said to be permutations, whereas the order in which they are represented is called combination. Both concepts are very important in Mathematics. 

What is Permutation?

In mathematics, permutation relates to the act of arranging all the members of a set into some sequence or order. In other words, if the set is already ordered, then the rearranging of its elements is called the process of permuting. Permutations occur, in more or less prominent ways, in almost every area of mathematics. They often arise when different orderings on certain finite sets are considered.

What is a Combination?

The combination is a way of selecting items from a collection, such that (unlike permutations) the order of selection does not matter. In smaller cases, it is possible to count the number of combinations. Combination refers to the combination of n things taken k at a time without repetition. To refer to combinations in which repetition is allowed, the terms k-selection or k-combination with repetition are often used. Permutation and Combination Class 11 is one of the important topics which helps in scoring well in Board Exams.

Permutation and Combination Formulas

There are many formulas involved in permutation and combination concepts. The two key formulas are:

Permutation Formula

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)!

Combination Formula

A combination is the choice of r things from a set of n things without replacement and where order does not matter.

Learn how to calculate the factorial of numbers here.

Difference Between Permutation and Combination

Go through the differences between permutation and combination given below.

Permutation	Combination
Arranging people, digits, numbers, alphabets, letters, and colours	Selection of menu, food, clothes, subjects, team.
Picking a team captain, pitcher and shortstop from a group.	Picking three team members from a group.
Picking two favourite colours, in order, from a colour brochure.	Picking two colours from a colour brochure.
Picking first, second and third place winners.	Picking three winners.

Uses of Permutation and Combination

A permutation is used for the list of data (where the order of the data matters) and the combination is used for a group of data (where the order of data doesn’t matter).

Solved Examples of Permutation and Combinations

Example 1: 

Find the number of permutations and combinations if n = 12 and r = 2.

Solution: 

Given,

n = 12
r = 2

Using the formula given above:

Permutation:

nPr = (n!) / (n-r)! =(12!) / (12-2)! = 12! / 10! = (12 x 11 x 10! )/ 10! = 132

Combination:

\(\begin{array}{l}_{n}C_{r} = \frac{n!}{r!(n-r)!}\end{array} \)

\(\begin{array}{l}\frac{12!}{2!(12-2)!} = \frac{12!}{2!(10)!} = \frac{12\times 11\times 10!}{2!(10)!} = 66\end{array} \)

Example 2:

In a dictionary, if all permutations of the letters of the word AGAIN are arranged in an order. What is the 49th word?

Solution:

Start with the letter A	The arranging the other 4 letters: G, A, I, N = 4! = 24	First 24 words
Start with the letter G	arrange A, A, I and N in different ways: 4!/2!  = 12	Next 12 words
Start with the letter I	arrange A, A, G and N in different ways: 4!/2! = 12	Next 12 words

This accounts up to the 48th word. The 49th word is “NAAGI”.

Example 3:

In how many ways a committee consisting of 5 men and 3 women, can be chosen from 9 men and 12 women?

Solution:

Choose 5 men out of 9 men = 9C5 ways = 126 ways

Choose 3 women out of 12 women = 12C3 ways = 220 ways

Total number of ways = (126 x 220)= 27720 ways

The committee can be chosen in 27720 ways.

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