Question

Prove that:

0! = 1

Class 11

Permutations and Combinations​

Answer 1

Answer:

$0! = 1 \\ \\ take \\ \\ n! = n(n - 1)! \\ \\ put \: n = 1 \\ \\ 1! = 1(1 - 1)! \\ \\ 1 = 1(0)! \\ \\ 1 ! = 0 !\\ \\ 1 = 0! \\ \\ hence \: proved.$

Answer 2

SOLUTION :-

Let n be a whole number, where n! is defined as the product of factors including n itself and everything below it. What it means is that you first start writing the whole number nn then count down until you reach the whole number 11.

●The general formula of factorial can be written in fully expanded form as,

》 n! = n×(n-1)×(n-2)×......×3×2×1

●or in partially expanded form as,

》n! = n×(n-1)!

●We know with absolute certainty that 1!=1, where n = 1. If we substitute that value of n into the second formula which is the partially expanded form of n!, we obtain the following:-

》n! =n×(n-1)!

》1! = 1×(1-1)!

》1!= 1× (0)!

》1! = 1 × 0!

●For the above equation to be true, we must force the value of zero factorial to equal 1, and no other. Otherwise, 1!≠1 which is a contradiction.

☆ So yes, 0! = 1 is correct because mathematicians agreed to define it that way (nothing more and nothing less) in order to be consistent with the rest of mathematics.

HENCE PROVED

HOPE IT HELPS :)