Question

What is the formula of $\begin{array}{rcl}& & {}_n{}^{n}C\end{array}$?

Answer 1

Find the formula for ${}_n{C}^n$.

in the formula: ${}_{\mathbf{r}}{}^{\mathbf{n}}\mathit{C}\mathbf{=}\frac{\mathbf{n}\mathbf{!}}{\mathbf{r}\mathbf{!}\left(n-r\right)\mathbf{!}}$

Replace $r$ with $n$ in the above formula:

$\begin{array}{rcl}{}_n{}^{n}C& =& \frac{n!}{n!\left(n-n\right)!}\\ & \Rightarrow & {}_n{}^{n}C=\frac{n!}{n!\left(0\right)!}\\ & \Rightarrow & {}_n{}^{n}C=\frac{1}{\left(0\right)!}\\ & \Rightarrow & {}_n{}^{n}C=1[\left(0\right)!=1]\end{array}$

Hence, the formula is $\begin{array}{rcl}{}_n{}^{n}C& =& 1\end{array}$.