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There are several explanations that I found interesting: The first one is to preserve the identity (n+1)!=(n+1)×n!. When we take n=0, we have 1!=1×0! which implies that 0!=1. The second one, which I believe is the real interpretation and intuition behind the notion of factorials, is that n! represents the number of permutations of n objects, let's say elements of the set {1,2,...,n}. Now suppose that you have 0 object, which is in the empty set {}. In how many ways you can permute the object? Only one; leave the set alone. Hence 0!=1.
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