0!=1?

Let n be any integer

so, $n!$ = $1\times2\times3\times\dots\times (n-1)\times n$

and, $n-1!$ = $1\times2\times3\times\dots\times (n-1)$

now, multiplying and dividing R.H.S. of $n-1!$ by n, we get

$\frac { 1\times 2\times 3\times ....\times n-1\times n }{ n }$

which is actually $\frac { n! }{ n }$

now, put $n=1$ in $n-1!$ , we get

$1-1!$ = $0!$ on L.H.S. and $\frac { 1! }{ 1 }$ = $1$ in R.H.S.

so, $0!$ = $\boxed1$

Please give some more solutions to this..Thank you!

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Comments

If n! is defined as the product of all positive integers from 1 to n, then: 1! = 1*1 = 1

2! = 1*2 = 2

3! = 123 = 6

4! = 123*4 = 24 ...

n! = 123...(n-2)(n-1)n

and so on.

Logically, n! can also be expressed n*(n-1)! .

Therefore, at n=1, using n! = $n*(n-1)!$

1! = 1*0! which simplifies to 1 = 0!

A Former Brilliant Member - 5 years, 11 months ago

The most comprehensible explanation by far...! Thank you :)

Dhruv Saxena - 5 years, 7 months ago

You are welcome...

A Former Brilliant Member - 5 years, 7 months ago

Using words alone you can determine 0!=1. factorials are used to tell us how many ways there are to arrange n amount of objects. if there are 0 objects there is only 1 way to arrange it. The proof is a nice way to show it but for basic understanding of what a factorial is and does words can help people who are learning it for the first time.

Preston Kilian - 5 years, 11 months ago

One can prove this using definition of permutations.

shivamani patil - 5 years, 11 months ago

Could you expand it?

Yoogottam Khandelwal - 5 years, 11 months ago

Very good explanation of 0!=1 by Yoogottam Khandelwal.

Jamil Osman - 5 years, 11 months ago

Thank you!

Yoogottam Khandelwal - 5 years, 11 months ago

niceee....;)

jarin tasnim - 5 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$