Nib asked
Why is zero factorial equal to one?
0! = 1 because there is exactly one way to line up nothing, and every rule that works for other factorials keeps working only if 0! is 1.
5 steps, drawn by Nib. Read it here, or watch him draw it. Updated .
1line-ups
The factorial n! is the number of different orders in which n distinct things can be lined up. For three letters, A, B and C, there are six orders, and 3! = 3 × 2 × 1 = 6. A common guess is that 0! should be 0, because with nothing to arrange there is nothing to count. That guess is the myth this page clears up.
n! counts the ways to line up n things.
a common mix-up0! is not 0: with nothing to arrange there is still one way, the empty line-up.
2zero things
Two things, A and B, line up as AB or BA, so 2! = 2. One thing has one order, so 1! = 1. With no things at all there is exactly one way to arrange them: the empty arrangement, where you do nothing. Counting that single way gives 0! = 1.
There is exactly one way to arrange nothing, so 0! = 1.
How many ways can zero things be lined up?
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One: the empty line-up, doing nothing.
3the ladder
Since n! = n × (n−1)!, you can step down by dividing: (n−1)! = n! ÷ n. Starting from 4! = 24, you get 3! = 6, 2! = 2, 1! = 1 and then 0! = 1 ÷ 1 = 1. One more step down would need dividing by 0, so the ladder stops exactly at 0!, and the pattern puts 1 there.
Stepping down by n! ÷ n lands on 1 at zero.
What stops the ladder going below 0!?
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The next step down would divide by zero, which is impossible.
4choosing
The number of ways to choose k things from n is n! ÷ (k! × (n−k)!). Choosing all 3 of 3 friends can only happen one way, but the formula gives 3! ÷ (3! × 0!) = 1 ÷ 0!. For the formula to give the true answer of 1, 0! has to be 1.
The choosing formula only works at the edges if 0! = 1.
5one forced value
Zero factorial is a definition, but not an arbitrary one: counting line-ups, the divide-down ladder and the choosing formula all demand 1. A smooth curve through the points (n, n!), the gamma function, agrees: it equals n! at whole numbers, and its value at 1 on the gamma scale is exactly 1, which is 0!.
0! = 1 is the one value that keeps every factorial rule working.
Zero factorial is one because there is exactly one way to arrange nothing, and every factorial rule insists on it.