What is a factorial?
The factorial of n, written n! and read “n factorial”, is the product of every whole number from 1 up to n. It is the number of ways to put n different things in order: three books can stand on a shelf in 3! = 6 ways. 0! is 1, by definition. Factorials grow very fast: 52!, the number of ways to shuffle a deck of cards, has 68 digits.
Source OpenStax Contemporary Mathematics · 7.2 Permutations: factorials, 0! = 1 and 52! ↗
How to calculate a factorial
Here it is with your numbers:
- 10! = 10 × 9 × 8 × … × 2 × 1
- 10 × 9 is 90
- 90 × 8 is 720
- 720 × 7 is 5,040
- 5,040 × 6 is 30,240
- 30,240 × 5 is 151,200
- 151,200 × 4 is 604,800
- 604,800 × 3 is 1,814,400
- 1,814,400 × 2 is 3,628,800
- 10! = 3,628,800
It means there are 3,628,800 ways to put 10 different things in order.
Factorial table
The factorials from 0! to 20!. Tap one to see the steps.
| n | n! |
|---|---|
| 0! | 1 |
| 1! | 1 |
| 2! | 2 |
| 3! | 6 |
| 4! | 24 |
| 5! | 120 |
| 6! | 720 |
| 7! | 5,040 |
| 8! | 40,320 |
| 9! | 362,880 |
| 10! | 3,628,800 |
| 11! | 39,916,800 |
| 12! | 479,001,600 |
| 13! | 6,227,020,800 |
| 14! | 87,178,291,200 |
| 15! | 1,307,674,368,000 |
| 16! | 20,922,789,888,000 |
| 17! | 355,687,428,096,000 |
| 18! | 6,402,373,705,728,000 |
| 19! | 121,645,100,408,832,000 |
| 20! | 2,432,902,008,176,640,000 |
FAQs
What is a factorial?
The factorial of a whole number n, written n!, is the product of every whole number from 1 up to n. 4! is 4 × 3 × 2 × 1, which is 24. It is the number of ways to put n different things in order.
Source OpenStax Contemporary Mathematics · 7.2 Permutations: factorials, 0! = 1 and 52! ↗
Why is 0 factorial 1?
0! is defined to be 1 so that the counting formulas work. The number of ways to order all n of n things is n! ÷ (n − n)!, which is n! ÷ 0!; that only gives the right answer, n!, if 0! is 1.
What is 52 factorial?
52! is 8.0658 × 10⁶⁷, a number with 68 digits. It is the number of different ways to shuffle a deck of 52 cards.
What is 10 factorial?
10! is 3,628,800: 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1.
What is the factorial of 100?
100! is 9.3326 × 10¹⁵⁷. Written out in full it has 158 digits, ending in 24 zeros; type 100 in the calculator to see them all.