Factorial
See how to solve a factorial n!
(+word examples).
4 examples and their solutions.
Factorial
Formula
n! = n⋅(n - 1)⋅(n - 2)⋅ ... ⋅3⋅2⋅1
Multiply from n to 1. Meaning: Pick & arrange n things
1! = 1
0! = 1
0! is defined as 1. 0! = 1
Example
5!
Solution 5! = 5⋅4⋅3⋅2⋅1
= 20⋅6
= 120
= 20⋅6
= 120
Close
Example
7!4!
Solution 7!4!
= 7⋅6⋅5⋅4!4!
= 7⋅6⋅5 - [1]
= 7⋅30
= 210
= 7⋅6⋅5⋅4!4!
= 7⋅6⋅5 - [1]
= 7⋅30
= 210
[1]
7! = 7⋅6⋅5⋅4⋅3⋅2⋅1
= 7⋅6⋅5⋅4!
To cancel the denominator 4! easily,
7! = 7⋅6⋅5⋅4!.
= 7⋅6⋅5⋅4!
To cancel the denominator 4! easily,
7! = 7⋅6⋅5⋅4!.
Close
Example
4 letters: a, b, c, d
Find the number of ways to arrange the letters.
Solution Find the number of ways to arrange the letters.
a, b, c, d
N = 4! - [1]
= 4⋅3⋅2⋅1
= 4⋅6
= 24
N = 4! - [1]
= 4⋅3⋅2⋅1
= 4⋅6
= 24
[1]
Pick and arrange 4 letters.
→ 4!
→ 4!
Close
Example
3 letters: a, b, c
4 numbers: 1, 2, 3, 4
Find the number of ways to arrange the letters and the numbers
when all letters are adjacent to each other.
Solution 4 numbers: 1, 2, 3, 4
Find the number of ways to arrange the letters and the numbers
when all letters are adjacent to each other.
a, b, c, 1, 2, 3, 4 - [1]
N = 5!⋅3! - [2]
= 5⋅4⋅3⋅2⋅1⋅32⋅1
= 20⋅6⋅6
= 20⋅36
= 720
N = 5!⋅3! - [2]
= 5⋅4⋅3⋅2⋅1⋅32⋅1
= 20⋅6⋅6
= 20⋅36
= 720
[1]
All letters are adjacent to each other.
→ Group the letters.
→ Group the letters.
[2]
Group, 1, 2, 3, 4
→ Pick and arrange 5 things.
→ 5!
For each case,
the letters (a, b, c) can be arranged in the Group.
→ Pick and arrange 3 things.
→ × 3!
Number of Ways (Math)
→ Pick and arrange 5 things.
→ 5!
For each case,
the letters (a, b, c) can be arranged in the Group.
→ Pick and arrange 3 things.
→ × 3!
Number of Ways (Math)
Close