Charts & references · 3 min read
Factorials Chart: 0! to 20! and What They Count
The chart
| n | n! | n | n! |
|---|---|---|---|
| 0 | 1 | 11 | 39,916,800 |
| 1 | 1 | 12 | 479,001,600 |
| 2 | 2 | 13 | 6,227,020,800 |
| 3 | 6 | 14 | 87,178,291,200 |
| 4 | 24 | 15 | 1,307,674,368,000 |
| 5 | 120 | 16 | 20,922,789,888,000 |
| 6 | 720 | 17 | 355,687,428,096,000 |
| 7 | 5,040 | 18 | 6,402,373,705,728,000 |
| 8 | 40,320 | 19 | 121,645,100,408,832,000 |
| 9 | 362,880 | 20 | 2,432,902,008,176,640,000 |
| 10 | 3,628,800 |
What n! counts
n! is the number of ways to arrange n different objects in a row. Five books can be lined up in 5! = 120 orders.
That is why factorials appear in permutations and combinations.
Why 0! = 1
There is exactly one way to arrange nothing, and defining 0! = 1 keeps formulas like nCr = n!/(r!(n−r)!) working at the edges.
15! with exact arithmetic
- 13 digits
- 3 trailing zeros
Step-by-step working 3 steps
- 1Definition
n! = n × (n − 1) × … × 2 × 1, and 0! = 1
The factorial counts the ways to arrange n different items in a row.
- 2Multiply it out
15! has 13 digits
The product is computed with exact whole-number arithmetic, so no digits are rounded.
- 3Trailing zeros
Count the factors of 5: ⌊15/5⌋ = 3
Each factor of 5 pairs with a factor of 2 to make a trailing zero.
Growth
Factorials outgrow exponentials: 10! = 3,628,800 is already larger than 2²⁰ = 1,048,576. The factorial calculator handles values up to 1000!.
Frequently asked questions
What is 5 factorial?
5! = 5 × 4 × 3 × 2 × 1 = 120.
Can you take the factorial of a decimal?
Not with the ordinary definition; the gamma function extends it to non-integers.
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