Question

Any Aryabhatta's brain here?

Question:-
I happened to come across 0! (Factorial of 0) where I got the result as 1. How?
In other words, $\tt \green{\: Prove \rarr} \: \orange{0!=1}$
​

Answer 1

A zero factorial is a mathematical expression for the number of ways to arrange a data set with no values in it, which equals one. In general, the factorial of a number is a shorthand way to write a multiplication expression wherein the number is multiplied by each number less than it but greater than zero. 4! = 24, for example, is the same as writing 4 x 3 x 2 x 1 = 24, but one uses an exclamation mark to the right of the factorial number (four) to express the same equation

Answer 2

Step-by-step explanation:

Given :-

0!

To find :-

Prove that 0! = 1

Solution :-

We know that

Number of ways of SELECTING ‘r’ objects from ’n’ objects n c r =n! / r!(n-r)! -------(1)

if we put n=1 and r = 1 then (1) becomes

1 c1 = 1! /1! (1–1)!

=> 1 = 1 /(1 * 0!)

Since one object can be selected from 1 object in only 1 way

=> 1 = 1 / 0!

=>1 × 0! = 1

=> 0 ! = 1/1

=> 0! = 1

Hence, Proved.

Another way :-

We know that

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

=> n ! = The product of all positive integers less than or equal to n

So, The value of 0! is 1, according to the convention for an empty product.

let consider

3! = 3×2×1

=> 3! = (4×3×2×1)/4

=> 3! = 4!/4

and

2! = 2×1

=> 2! = (3×2×1)/3

=> 2! = 3!/3

and

1! = 1×2/2

=> 1! = 2!/2

If you observe the pattern

we get n! = (n+1)!/n----(1)

Put n = 0 then

=> 0! = (0+1)/1

=> 0! = 1/1

=> 0! = 1

Hence, Proved.

Answer :-

0! = 1

Used formulae:-

• n! = n(n-1)(n-2)...1