Two shortest ways to prove that 0!=1 zero factorial

Prove that 0!=1 – (Zero Factorial equals number one)

Way I:
Note 0!=1
proof zero factorial

Way II:

If you wanna skip this permutation short explanation, jump to the bold word ‘Concept’ below.

Permutation
Number of permutations of ‘n‘ different things taken ‘r‘ at a time is given by

basic permutation

Proof: Say we have ‘n‘ different things.
Clearly the first place can be filled up in ‘n‘ ways. Number of things left after filling-up the first place =n-1.
So the second-place can be filled-up in (n-1) ways. Now number of things left after filling-up the first and second places =n-2
Now the third place can be filled-up in (n-2) ways.
Thus number of ways of filling-up first-place =n
Number of ways of filling-up second-place =n-1
Number of ways of filling-up third-place =n-2
Number of ways of filling-up r-th place =n-(r-1)=nr+1
By multiplication – rule of counting, total number of ways of filling up, first, second, … , r-th place together:
=n(n-1)(n-2)·…·(nr+1)

Hence:
n things taken r

Number of permutations of ‘n‘ different things taken all at a time is given by
nPn=n!

Proof:
Now we have ‘n‘ objects, and n-places.
Number of ways of filling-up first-place =n
Number of ways of filling-up second-place =n-1
Number of ways of filling-up third-place =n-2
Number of ways of filling-up r-th place, i.e. last place =1
Number of ways of filling-up first, second, …, n-th place
=n(n-1)(n-2)·…·3·2·1
nPn=n!

Concept: We have

basic permutation

Putting r=n, we have
nPn=n!/0!

But nPn=n!
Clearly it is possible, only when 0!=1. Hence it is proof that 0!=1.
Note: Factorial of negative-number is not defined. The expression (–1)! has no meaning.

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