# How can the factorial of 0 be 1?

Jun 10, 2015

If you know the value of n! then you can calculate (n-1)! as
(n!)/n; since 1! =1 then 0! = (1-1)! = 1/1 = 1

#### Explanation:

Actually Nelson's answer if probably correct, but there is some justification for the definition.

Jun 10, 2015

Because there is one permutation of zero objects.

#### Explanation:

Interpreting n! to be the number of permutations of $n$ objects.

And agreeing that there is a permutation of $0$ object (namely the empty permutation).

Leads one to state that 0! = 1

Jun 11, 2015

We can do some MATH and STUFF!

At ${n}_{0} = 1$:
(n!)/((n+1)!) = (1*cancel(2*3*4*...*n))/(cancel(2*3*4*5*...*n*)(n+1))

$= \frac{1}{n + 1}$

If (n!)/((n+1)!) = 1/(n+1) = (0!)/(1!), then, with $n = 0$:

0! = 1!*(1/(0+1)) = (1!)/(1) = 1