I am looking for a quick and easy explanation as to why 0! is 1. I can create the proof and illustrate in the proof that zero can not be in the denominator, but this is difficult for my students to understand. I am a secondary ed. teacher. My name is Donna. Thanks for your help!
There is no 'proof' that 0! is 1, it is a convention that we adopt for convenience. When we develop the number of ways of choosing k of n distinct objects, nCk, we end up with n!/(k!(n-k)!). If we were to allow k=0 in this we would want the answer to be 1, in how many ways can we choose none of the objects - there's just one thing to do, leave them all. Thus we would like n!/0!n! = 1, and we define 0! = 1.
A convincing explanation of 0! is to think of n! as the number of ways to make an ordered list out of n objects.
In that spirit 0! should represent the number of ways to make an ordered list with 0 objects. There is exactly one way - leave the page blank!
More dramatically, think of the number of ways to pay a toll with n distinct coins. That is n!
Now how do you pay a toll with no coins - only one way - drive on through!
Denis and Walter
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