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POFactorial means “keep multiplying down by whole numbers”. So four factorial means four times three times two times one. The key pattern is that any factorial can be rewritten as the number itself multiplied by the factorial just before it: four factorial equals four times three factorial, five factorial equals five times four factorial, and in general n factorial equals n times the previous factorial.
That is why zero factorial equals one. If one factorial equals one times zero factorial, and one factorial is already one, then zero factorial must be one for the pattern to stay consistent. It also matches the combinatorics idea: factorial counts arrangements, and the number of ways to arrange nothing is one — the single “empty” arrangement.
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