Factorial of 0
0! = 1
1 digits
How it is calculated
0! = 1 = 1
Facts about 0!
| Digits | 1 |
|---|---|
| Trailing zeros | 0 (one for every factor 5 × 2) |
| Stirling estimate | 1 |
| 0! ÷ 0! | 1 |
Nearby factorials
| 1! | 1 |
| 2! | 2 |
| 5! | 120 |
| 10! | 3628800 |
What are factorials used for?
Counting arrangements: 0! is the number of different orders in which 0 distinct items can be lined up. Factorials also appear in probability, combinations and series.
Counting with 0!
There are 1 different ways to arrange 0 different books on a shelf: 0 choices for the first place, 0 for the second, and so on down to one choice for the last.
In probability, if 0 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 1. That is why shuffled decks and passwords with many symbols are so hard to guess.
How big is 0!?
0! has 1 digits. The digits add up to 1. Stirling's formula, √(2πn) × (n/e)^n, gives a close estimate without multiplying everything out, and gets more accurate as n grows.
Shuffles where nothing stays put
A related count is the number of derangements of 0 items: orders in which no item ends up in its original place, as in a Secret Santa where nobody draws their own name. For 0 items there is exactly 1, which is 100% of all 1 orders. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.
Computing 0! yourself
In Excel or Google Sheets, type =FACT(0). Spreadsheets keep only 15 significant digits, so from 18! on the result is rounded, and FACT stops at 170!, the largest factorial that fits in a standard floating-point number. In Python, math.factorial(0) returns the exact integer with all 1 digits, because Python integers have no size limit. The gamma function extends factorials to fractions: Γ(n + 1) = n!, so (1/2)! = √π ÷ 2 ≈ 0.8862.
Why is 0! equal to 1?
There is exactly one way to arrange nothing, and the rule n! = n × (n − 1)! only works for n = 1 if 0! = 1. Defining it this way keeps formulas for combinations and series correct at the edges.
How this answer was calculated
The result is worked out from the numbers in this page's address when the page is opened, using exact arithmetic wherever possible and rounding only for display. Nothing is copied from a printed table. If you think an answer is wrong, please tell us through the contact page and we will check it.
Reference this page
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"Factorial of 0". Sumwise. https://sumwisemath.com/calc/factorials/factorial-of-0/
Questions
What is 0 factorial?
0! is the product of every whole number from 1 to 0 (0! is defined as 1).
How many digits does 0! have?
1. The number of digits of n! is the whole part of log10(n!) plus one.
How many zeros are at the end of 0!?
0. Count the factors of 5 in the numbers 1 to 0: there are always more factors of 2, so every 5 makes one trailing zero.
Is 0! divisible by 1?
Yes, 0! = 1 is divisible by 1.
What is the last non-zero digit of 0!?
1. Strip the 0 trailing zeros and look at the digit that is left at the end.
What is 1! in terms of 0!?
1! = 1 × 0!. Each factorial is the previous one multiplied by the next whole number.