Permutation and Combination

What are Permutation and Combination?

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.

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.

${}_n{P}_r=\frac{n!}{(n-r)!}$

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

Permutation and Combination Problems

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:

${}_n{P}_r=\frac{n!}{(n-r)!}$

$=\frac{12!}{(12-2)!}$

$=\frac{12!}{\left(10\right)!}=\frac{12\times 11\times 10}{\left(10\right)!}=132$

Combination:

${}_n{C}_r=\frac{n1}{r!(n-r)!}$

$\frac{12!}{2!(12-2)!}$

$=\frac{12!}{2!\left(10\right)!}=\frac{12\times 11\times 10}{2!\left(10\right)!}=66$

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 like the NCERT solutions for Permutations and Combinations.