Question

How do you calculate \[6!\]?

Answer 1

Hint: A factorial is a number multiplied by all the integers less than the number. It is denoted as \[n!\]. For a number n, the factorial of this number n is \[n!=n\times n-1\times n-2\times ......\times 3\times 2\times 1\]. Factorials of 0 and 1 are both equal to 1. We use this factorial concept in various areas of mathematics like in probability, Taylor series, binomial expansion, etc.

Complete step by step answer:
As per the given question, we are provided with a number and we are asked to find the factorial value of this number. And the given number is 6.
As the number 6 is greater than 0, we can say that this is a positive integer.
We know that for any natural number n, the factorial value of n is \[n!\] which is obtained by multiplying n with all the integers less than this number till 0. That is,
\[n!=n\times n-1\times n-2\times ......\times 3\times 2\times 1\].
All the integers less than 6 are 1, 2, 3, 4 and 5.
So, the factorial of 6 will be equal to the multiplication of integers from 1 to 6. That we can write as
\[\Rightarrow 6!=6\times 5\times 4\times 3\times 2\times 1\]
Starting from the right, \[1\times 2\] is 2 which when multiplied by 3 gives 6 \[\Rightarrow 2\times 3=6\]. Now, when this 6 is multiplied by 4, we get 24 \[\Rightarrow 6\times 4=24\]. And, multiplication of 24 by 5 gives 120 \[\Rightarrow 24\times 5=120\]. At last, when 120 is multiplied by 6, we get 720 \[\Rightarrow 120\times 6=720\].

\[\therefore 6!=720\] is the required answer.

Note: In order to solve these types of problems, we need to have proper knowledge on the concept of factorials. There is a special function called Euler’s gamma function which can be used to find the factorial of negative integers. We should pay attention in the calculation part where generally the answer goes wrong. We need to avoid calculation mistakes to get the expected results.