Question

What is a factorial of $0$?

Answer 1

Meaning of factorial:

• Factorial is the product of positive numbers, each number is one less than the preceding number.
• Factorial is denoted by $!$
• $\mathbf{n}\mathbf{!}\mathbf{=}\mathbf{n}\mathbf{\times }\left(n-1\right)\mathbf{\times }\left(n-2\right)\mathbf{\times }\left(n-3\right)\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{3}\mathbf{\times }\mathbf{2}\mathbf{\times }\mathbf{1}$
• The factorial of zero is $1$, symbolically represented as $0!=1$
• $$\begin{gathered} {}^{\mathbf{n}}\mathit{C}_{\mathbf{r}}\mathbf{=}\frac{\mathbf{n}\mathbf{!}}{\mathbf{r}\mathbf{!}\mathbf{\times }\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{r}\mathbf{)}\mathbf{!}} \\ \text{Put}r=n \\ {}^{n}C_n=\frac{n!}{n!\times (n-n)!}=\frac{1}{0!}\mathbf{[}\mathbf{\because }{}^{\mathbf{n}}\mathbf{C}_{\mathbf{n}}\mathbf{=}\mathbf{1}\mathbf{]} \\ \Rightarrow 1=\frac{1}{0!} \\ \Rightarrow 0!=1 \end{gathered}$$

Hence, the value of factorial $0$ is $1$.