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n factorial is product of all numbers between n and 1.

0 factorial is (0 * 1 = 0).

Why is 0 factorial 1?

How can I proof this in mathematical way?

User
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  • Factorial is about permutation without repetitions of $n$ objects.

    Imagine you have $0$ objects.

    How many possible permutations of those $0$ objects can you make? Just one.

    $$0! = 1$$

     – Enrico M. Jan 28 '16 at 12:53
  • I understood. Thank you for answering. – User Jan 28 '16 at 12:54

1 Answers1

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It is just for definition of ! function 0=1! And n!=(n-1)!n

Giovanni Siclari
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  • It's not just "by definition".

    Factorial is about permutation without repetitions of $n$ objects.

    Imagine you have $0$ objects.

    How many possible permutations of those $0$ objects can you make? Just one.

    $$0! = 1$$

     – Enrico M. Jan 28 '16 at 12:53
  • However, there just happens to be another reason why 0 factorial is equal to 1, aside from the fact that there is one possible permutation for zero. Suppose that we had n objects, for which we wished to permute into n number of places. From the permutation formula, we could deduce that the number of permutations for n objects into n places would equal n!/0!. On the other hand, we could interpret, from elementary deductive logic, that the result would eventually be 1/n. Thus, for n!/0! to be equivalent to 1/n, the only logical solution is to allow 0 factorial to be 1. – LÜHECCHEgon Apr 20 '19 at 07:39
  • https://www.khanacademy.org/math/precalculus/prob-comb/combinatorics-precalc/v/zero-factorial-or-0 – LÜHECCHEgon Apr 20 '19 at 07:39