Permutations (nPr)

Permutations (nPr)

How to solve permutation problems (nPr): formulas, examples, and their solutions.

Formula: nPr

nPr is starting from n and multiplying r factors.

nPr 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.

nPr = n!/(n - r)!

This formula is used
when proving formulas or properties of
permutations or combinations.

Factorial (n!)

Combinations (nCr)

Example 1

Find the value of the given number. 6P4

6P4 is
starting from 6 and multiplying 4 factors.

So 6P4 = 6⋅5⋅4⋅3.

6⋅5 = 30
4⋅3 = 12

30⋅12 = 360

So 360 is the answer.

Example 2

There are 9 students in a class. Find the number of ways to choose 4 students and arrange them in a row.

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
9P4.

9P4 is
starting from 9 and multiplying 4 factors.

So 9P4 = 9⋅8⋅7⋅6.

9⋅8 = 72
7⋅6 = 42

72⋅42 = 3024

So 3024 is the answer.

Example 3

Numbers {1, 2, 3, 4, 5, 6, 7, 8} are given. By using the numbers once, find the number of ways to make a 3-digit number.

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
8P3.

8P3 is
starting from 8 and multiplying 3 factors.

So 8P3 = 8⋅7⋅6.

7⋅6 = 42

8⋅42 = 336

So 336 is the answer.

Formula: nPn

nPn = n!

nPn = n!

So when the former and latter numbers of P
are the same,
simply change the permutation to factorial.

Factorial (n!)

Example 4

Find the value of the given number. 5P5

5P5 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: nP1

nP1 = n

nP1 is
starting from n and multiplying 1 factor: itself.

So nP1 = n.

Example 5

Find the value of the given number. 9P1

The latter number of 9P1 is 1.

So 9P1 = 9.

Formula: nP0

nP0 = 1

nP0 is defined as 1.

nP0 means
from n things,
[choose] 0 things
and arrange them [in order].

There's only 1 way to do that:
not choosing anything.

So nP0 = 1.

Example 6

Find the value of the given number. 2P0

The latter number of 2P0 is 0.

So 2P0 = 1.