Permutation and combination is 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 about the related concepts with a diverse set of solved questions. Moreover, practice questions improve your skills and help you solve any question at your own pace.

**Table of Contents:**

- Permutation and Combination Definition
- Difference Between Permutation and Combination
- Formulas
- Problems
- Practice Questions

## Permutation and Combination Definition

In mathematics, the notion of 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.

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.

Combinations refer 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.

**Standard Definition of Permutation and Combination**

Permutation: Permutation means an arrangement of things. The word arrangement is used if the order of things is considered.

Combination: Combination means selection of things. The word selection is used, when the order of things has no importance.

### Difference Between Permutation and Combination

Permutation |
Combination |

Arranging people, digits, numbers, alphabets, letters, and colors | 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 favorite colors, in order, from a color brochure. | Picking two colors from a color brochure. |

Picking first, second and third place winners. | Picking three winners. |

## Permutation and Combination Formulas

## 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.

### 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

## Practice 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 about 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.