Why is the factorial of 0 equal to 1?

Question

Wouldn’t this also mean that 1!=0!, why is this true?

In: 383

Answer 1

Others have explained the 0! = “number of ways to rearrange 0 people” part, so I’ll try to convince you that 0! shouldn’t be 0.

You can think of 2 lines of people, with 2 distinct people in first one and 3 in the second.

How many distinct way can you rearrange them, assuming that no one leaves their line? Well, as for the first line, there are 2! arrangements. As for the second one, the answer is 3!.

Then, since you can pair 1 way of arranging the first line with 1 way of arranging the second line to make 1 distinct arrangement of the 2 lines, the number of distinct arrangements is 2! × 3!.

Now, what if the first line is empty? Then the answer should be 0! × 3!. If 0! = 0, this number should be 0. But you should get 3! different arrangements when shuffling the second line, no? Then adding an empty first line shouldn’t suddenly make the number of arrangement 0?

Look just how convenient it is for the number of permutation of an empty set to be 1!