Factorial
How to solve a factorial (n!): formula, 3 examples, and their solutions.
Formula
Formula
n! means
multiply from n to 1.
n! = n⋅(n - 1)⋅(n - 2) ... 3⋅2⋅1
[ ! ] is read as [factorial].
Special values:
1! = 1
0! = 1
Example 1
Example
Solution
5! is,
multiply from 5 to 1,
5⋅4⋅3⋅2⋅1.
5⋅4 = 20
3⋅2⋅1 = 6
20⋅6 = 120
So 120 is the answer.
Example 2
Example
Solution
The denominator is 4!.
So change 7! to 7⋅6⋅5⋅4!.
(7! = 7⋅6⋅5⋅4⋅3⋅2⋅1
= 7⋅6⋅5⋅4!)
Cancel both 4!.
6⋅5 = 30
7⋅30 = 210
So 210 is the answer.
Example 3
Example
Solution
There are 3 letters:
a, b, c.
And there are 4 numbers:
1, 2, 3, 4.
It says
all letters are adjacent to each other.
So group the letters.
So there are,
[group], 1, 2, 3, 4,
5 things to arrange.
n! is used
when finding the number of ways
to arrange n things.
So the number of ways
to arrange these 5 things is
5!.
For each case,
the letters [a, b, c] are also arranged.
So multiply the number of ways
to arrange these 3 letters:
3!.
Rule of Product
So the number of the ways is
5!⋅3!.
5! is,
multiply from 5 to 1,
5⋅4⋅3⋅2⋅1.
3! is,
multiply from 3 to 1,
3⋅2⋅1.
5⋅4 = 20
3⋅2⋅1 = 6
3⋅2⋅1 = 6
6⋅6 = 36
20⋅36 = 720
So 720 is the answer.