# Permutations

Permutations

Let us assume that you have 6 chairs and 4 people. If you randomly choose 4 chairs out of these 6, in how many ways can you arrange the 4 people? Well, rather than taking each case one by one, permutation helps us solve these types of questions.

Let us have a look at how different choices of arrangements can affect our total number of outcomes.

## Lesson Plan

 1 What do You Mean by Permutation 2 Important Notes on Permutation 3 Solved Examples on Permutation 4 Challenging Questions on Permutation 5 Interactive Questions on Permutation

## Definition of Permutation

Suppose you need to arrange n people in a row. How many distinct arrangements are possible?

We have n options to fill the first chair, $$\left( {n - 1} \right)$$ options to fill the second chair, $$\left( {n - 2} \right)$$ options to fill the third chair, and so on. Thus, using the FPC (Fundamental Principle of counting), the total number of arrangements of the n people in the row of n chairs is

$\left( n \right) \times \left( {n - 1} \right) \times \left( {n - 2} \right) \times ... \times 1 = n!$

Each particular arrangement is termed as a permutation of the n people. Hence, there will be a total of n! different permutations / arrangements.

## Formula of Permutation

Now, we introduce a new notation. The symbol $$^n{P_r}$$ is used to denote the number of permutations of n distinct objects, taken r at a time. Thus, permutation formula can be given as :

 $^n{P_r} = \frac{{n!}}{{\left( {n - r} \right)!}}$

Examples

\begin{align}&^5{P_2} = \frac{{5!}}{{\left( {5 - 2} \right)!}} = \frac{{5!}}{{3!}} = \frac{{120}}{6} = 20\\&^6{P_5} = \frac{{6!}}{{\left( {6 - 5} \right)!}} = \frac{{6!}}{{1!}} = \frac{{120}}{1} = 120\\&^7{P_2} = \frac{{7!}}{{\left( {7 - 2} \right)!}} = \frac{{7!}}{{5!}} = \frac{{7 \times 6 \times 5!}}{{5!}} = 42\end{align}

## Permutation Calculator

Enter the values of n and r in the permutation and combination calculator shown below to calculate the value of the permutations and combinations

## What are the types of Permutations

There are two types of permutations.

### Permutation with repetition

This formula is used to find the statistics of permutation (number of possible ways in which arrangement can be done) while allowing repetition.

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

### Permutation without repetition

This formula is used to find the statistics of permutation (number of possible ways in which arrangement can be done) while ensuring that there is no repetition.

 $$P\left ( n,r \right )$$ = $$n^{r}$$
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More Important Topics
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More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus
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