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QUESTION

Why is 0! (zero factorial) defined to be 1?

0! = 1 because it is an empty product.

Compare how we handle empty sums and empty products:

color(white)()Sums

The number 0 is the identity under addition, i.e.:

0+a = a+0 = a

for any number a.

Consider (the formula for triangular numbers):

T_n = sum_(k=0)^n k

If n=0, then the sum is empty and equal to the additive identity 0.

color(white)()Products

The number 1 is the identity under multiplication, i.e.:

1*a = a*1 = a

for any number a.

Consider (the formula for factorials):

n! = prod_(k=0)^n k

If n=0, then the product is empty and equal to the multiplicative identity 1.