What is the factorial of 0?

1 Answer
Aug 14, 2014

The simple answer is 1.

It is 1 by definition. However, this definition fits in nicely with applications for factorials.

Since you have posted this under binomial theorem, let's look at this application. For example: #(x+1)^3#.

Expanding this, we get: #1x^3+3x^2+3x+1#

The coefficients are determined by #1=_3 C_0#, #3=_3C_1#, #3=_3C_2#, #1=_3C_3#.

#._n C_r# is #n# choose #r# and defined as #(n!)/(n!(n-r)!#.

So, #._3 C_0=(3!)/(3!(3-0)!)=(3!)/(3!0!)#. If #0! =0#, then #._3 C_0# would be undefined. However, when we use #0! =1#, #._3 C_0=1# which gives us the proper coefficient. This explains why we want to define #0! =1#.