Consider one of the most basic uses of the factorial function - determining the number of permutations of n distinct objects. n distinct objects can be arranged in a line in n! ways. This is easy to see if you have 2 or more objects. If you have one, then you have no choice in how to arrange it, since 1! = 1. If you have no objects, you still have no choice in how to arrange it, thus 0! = 0. That's one explanation of why it's defined that way.
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Consider one of the most basic uses of the factorial function - determining the number of permutations of n distinct objects. n distinct objects can be arranged in a line in n! ways. This is easy to see if you have 2 or more objects. If you have one, then you have no choice in how to arrange it, since 1! = 1. If you have no objects, you still have no choice in how to arrange it, thus 0! = 0. That's one explanation of why it's defined that way.