Permutation and Combination: Permutation and Combination Formulas

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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

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:

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

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.