What is the value of zero factorial?

Zero factorial equals one, which looks like a trick until you see why it has to. A factorial counts the arrangements of a set: three items can be ordered in six ways, so three factorial is six. An empty set can be arranged in exactly one way, namely leaving it as it is, so zero factorial is one. The same result falls out of the recursive rule that n factorial equals n times the factorial of n minus one. Applying it to one gives one factorial equals one times zero factorial, and since one factorial is clearly one, zero factorial must be one. There is also a general convention that a product of no numbers at all equals one, just as a sum of no numbers equals zero, because one is the value that leaves multiplication unchanged. Defining it this way keeps formulas tidy, particularly the binomial coefficient, which would otherwise break whenever you chose nothing or everything.

One factorial also equals one, making zero and one the only inputs where the factorial matches its own argument closely enough to look coincidental.