Why does 0 factorial equal 1 instead of 0? Simple proof

Question

I understand that factorial means multiplying down:

5! = 5 \times 4 \times 3 \times 2 \times 1

and

3! = 3 \times 2 \times 1.

But then I see this:

0! = 1

and it feels wrong.

If there are zero numbers to multiply, why is the answer $1$ and not $0$ ? Is this just a definition mathematicians picked so formulas work, or is there an actual reason behind it?

I also saw someone say "there is one way to arrange nothing", but I do not fully get what that means.

Chen Wei · Accepted Answer

The cleanest reason is counting. n ! counts the number of ways to arrange n distinct objects. For example, if you have three objects, there are 3 ! = 6 possible arrangements. If you have one object, there is exactly one arrangement: 1 ! = 1. Now ask the same question for zero objects: How many ways are there to arrange nothing? There is exactly one way: leave the arrangement empty. That is why 0 ! = 1. It is not saying "zero times something equals one." It is saying the empty arrangement is one valid arrangement. This also keeps combinations correct: ( n 0 ) = n ! 0 ! ( n − 0 ) ! = 1. There is exactly one way to choose zero objects from any set: choose nothing.

Alex Kim · Answer

You can also get 0 ! = 1 by walking backward through the factorial pattern. For positive integers, n ! = n ( n − 1 ) ! . So: 3 ! = 3 ⋅ 2 ! 2 ! = 2 ⋅ 1 ! 1 ! = 1 ⋅ 0 ! . But 1 ! = 1, so the last line becomes 1 = 1 ⋅ 0 ! . Therefore 0 ! = 1. If we tried to set 0 ! = 0, the pattern would immediately break at 1 !. So it is a definition, but not an arbitrary one. It is the definition that preserves the counting meaning and the recursive factorial rule at the same time.

Emily Park · Answer

Another useful phrase is "empty product." The empty sum is 0 because 0 is the number that does not change addition: a + 0 = a . The empty product is 1 because 1 is the number that does not change multiplication: a × 1 = a . So when a product has no factors, the natural neutral value is 1, not 0. That idea is what makes many formulas behave cleanly. For example, in the binomial theorem: ( x + y ) n = ∑ k = 0 n ( n k ) x n − k y k . The edge cases k = 0 and k = n only work smoothly when 0 ! = 1. This is also why students sometimes confuse 0 ! with 0 0. They are related by "empty" reasoning in some contexts, but they are not the same expression.

Why does 0 factorial equal 1 instead of 0? Simple proof

3 Answers

Why does 0 factorial equal 1 instead of 0? Simple proof

3 Answers