Subject: why does 0! equal 1?

hi- this is Diane- I am a college student, but my statistics teacher asked us this question and I remember it being raised while I was in high school too...

Every math book always claims that 1!=1 and 0!=1 are givens, and that we should just memorize it. i understand the 1! part, but where is the basis for claiming that 0!=1???? thanks so much.

Hi Diane,

One way to see this is to think about n! as the ANSWER to a question: How many ways can you make an ordered list of n things (all different).

If THAT is you basic idea (and the formula nx(n-1)x .... 2x1 is the solution). THEN consider the question:

How many ways can you make a list of 1 thing : 1
How many ways can you make a list of 0 things : 1
It is quite possible to make a list of 0 things. A blank piece of paper is such a list. So there ARE ways to make the list - and they all look the same. One way.

This point of view would be called a 'combinatorial interpretation'.

It also fits with various other formulae which use n! for counting etc.

E.g. nCk (n choose k) is n!/k! (n-k)!. How about n C n? It should be 1 (only one way to choose n things form n things - chosse them all). It should also be  n!/0! n!. Those two answers match PROVIDED 0!=1.

Similarly n C 0 should be 1. (One way to choose nothing - just walk away). So we want 1 = n!/n! 0! Again that means we want 0!=1. It works so well, keeps so many formulae still working in the extreme cases that people accept it as the consistent value.


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