**Way I:**

Note 0!=1

**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

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)=

*n*–

*r*+1

By multiplication – rule of counting, total number of ways of filling up, first, second, … ,

*r*-th place together:

*n*(

*n*-1)(

*n*-2)·…·(

*n*–

*r*+1)

Hence:

Number of permutations of ‘

*n*‘ different things taken all at a time is given by

^{n}P

_{n}=

*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

^{n}P

_{n}=

*n*!

**Concept**: We have

Putting

*r*=

*n*, we have

^{n}P

_{n}=

*n*!/0!

But

^{n}P

_{n}=

*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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