# Permutations (_{n}P_{r})

How to solve permutation problems (_{n}P_{r}): formulas, examples, and their solutions.

## Formula: _{n}P_{r}

_{n}P_{r} is

starting from *n* and multiplying *r* factors.

When solving problems,

you don't have to memorize (*n* - *r* + 1).

Just start from *n* and multiply *r* factors.

_{n}P_{r} = *n*!/(*n* - *r*)!

This formula is used

when proving formulas or properties of

permutations or combinations.

Factorial (*n*!)

Combinations (_{n}C_{r})

## Example 1

_{6}P_{4} is

starting from 6 and multiplying 4 factors.

So _{6}P_{4} = 6⋅5⋅4⋅3.

6⋅5 = 30

4⋅3 = 12

30⋅12 = 360

So 360 is the answer.

## Example 2

It says

from 9 students,

choose 4 students,

and arrange them in a row.

Arrangement means [in order].

When there's [choosing] [in order],

then use the permutation.

So the number of ways to choose and arrange is_{9}P_{4}.

_{9}P_{4} is

starting from 9 and multiplying 4 factors.

So _{9}P_{4} = 9⋅8⋅7⋅6.

9⋅8 = 72

7⋅6 = 42

72⋅42 = 3024

So 3024 is the answer.

## Example 3

It says

from 8 numbers,

choose 3 numbers,

and make a 3-digit number.

Making a 3-digit number means [in order].

When there's [choosing] [in order],

then use the permutation.

So the number of ways to choose and arrange is_{8}P_{3}.

_{8}P_{3} is

starting from 8 and multiplying 3 factors.

So _{8}P_{3} = 8⋅7⋅6.

7⋅6 = 42

8⋅42 = 336

So 336 is the answer.

## Formula: _{n}P_{n}

_{n}P_{n} = *n*!

So when the former and latter numbers of P

are the same,

simply change the permutation to factorial.

Factorial (*n*!)

## Example 4

_{5}P_{5} is 5!.

5! is

multiplying from 5 to 1.

So 5! = 5⋅4⋅3⋅2⋅1.

Factorial (*n*!) - Example 1

5⋅4 = 20

3⋅2 = 6

20⋅6 = 120

So 120 is the answer.

## Formula: _{n}P_{1}

_{n}P_{1} is

starting from *n* and multiplying 1 factor: itself.

So _{n}P_{1} = *n*.

## Example 5

The latter number of _{9}P_{1} is 1.

So _{9}P_{1} = 9.

## Formula: _{n}P_{0}

_{n}P_{0} is defined as 1._{n}P_{0} means

from *n* things,

[choose] 0 things

and arrange them [in order].

There's only 1 way to do that:

not choosing anything.

So _{n}P_{0} = 1.

## Example 6

The latter number of _{2}P_{0} is 0.

So _{2}P_{0} = 1.