“Why does 0!=1?” zhongwei, my ex-student prompted me this question, scratching my rusty brain, i vaguely remembered n!= 1*2*3*…*n, but how does 0 comes in? read on if you want the answer …
Why does 0! = 1 ?
[source: Math Forum]
Usually n factorial is defined in the following way:
n! = 1*2*3*…*n
But this definition does not give a value for 0 factorial, so a natural question is: what is the value here of 0! ?
A first way to see that 0! = 1 is by working backward. We know that:
1! = 1
2! = 1!*2
2! = 2
3! = 2!*3
3! = 6
4! = 3!*4
4! = 24
We can turn this around:
4! = 24
3! = 4!/4
3! = 6
2! = 3!/3
2! = 2
1! = 2!/2
1! = 1
0! = 1!/1
0! = 1
In this way a reasonable value for 0! can be found.
How can we fit 0! = 1 into a definition for n! ? Let’s rewrite the usual definition with recurrence:
1! = 1
n! = n*(n-1)! for n > 1
Now it is simple to change the definition to include 0! :
0! = 1
n! = n*(n-1)! for n > 0
Why is it important to compute 0! ?