# Permutation and Combination: Permutation and Combination Formulas (For CBSE, ICSE, IAS, NET, NRA 2022)

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Permutation and combination are all about counting and arrangements made from a certain group of data.

## What is Permutation?

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.

## What is 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

## 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.

### 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 favorite 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 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) .

## Permutation and Combination Examples

Example 1: Find the number of permutations and combinations if and ?

Solution: Given,

Using the formula given above:

Permutation:

Combination:

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 First 24 words Start with the letter G Arrange A, A, I and N in different ways: Next 12 words Start with the letter I Arrange A, A, G and N in different ways: 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

Choose 3 women out of 12 women = ways

The committee can be chosen in ways.

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