Permutation And Combination

Permutation and combination are all about counting and arrangements made from a certain group of data. This is one of the most important topics in the list of mathematics. In this section, will discuss all the related concepts with a diverse set of solved questions along with permutation and combination formulas. Moreover, practice questions improve your skills and help you solve any question at your own pace.

Permutation Definition

In mathematics, permutation relates to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging its elements, a process called 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.

Combination Definition

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 Formulas

There are many formulas involved in permutation and combination concept. 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.

\( \large _{n}P_{r} = \frac{n!}{(n-r)!} \)

Combination Formula

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

Difference Between Permutation and Combination

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.

Permutation and Combination Problems

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:

\(_{n}P_{r} = \frac{n!}{(n-r)!}\) \(= \frac{12!}{(12-2)!} = \frac{12!}{(10)!} = \frac{12\times 11\times 10}{(10)!}=132\)

Combination:

\(_{n}C_{r} = \frac{n1}{r!(n-r)!}\) \(\frac{12!}{2!(12-2)!} = \frac{12!}{2!(10)!} = \frac{12\times 11\times 10}{2!(10)!} = 66\)

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! 1! 1! = 12 Next 12 words
Start with the letter I arrange A, A, G and N in different ways: 4!/2! 1! 1! = 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

The committee can be chosen in 27720 ways

Permutation and Combination Questions

Question 1: In how many ways can the letters be arranged so that all the vowels come together: Word is “IMPOSSIBLE”

Question 2: In how many ways of 4 girls and 7 boys, can be chosen out of 10 girls and 12 boys to make a team.

Question 3: How many words can be formed by 3 vowels and 6 consonants taken from 5 vowels and 10 consonants.

From the above discussion, students would have gained certain important aspects related to this topic. To gain further understanding of the topic, it would be advisable that students should work on sample questions with solved examples. To learn more about different maths concepts, register with BYJU’S today.

Practise This Question

If the HCF of 60 and 168 is 12, what is the LCM?

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