# 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 doesnŐšt 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 49 ^{th} 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 48^{th} word. The 49^{th} word is â€śNAAGIâ€ť .

**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.