Question

For which value of $r$, ${}^{n}P_r$ and ${}^{n}C_r$ are equal.

Answer 1

We know that,

${}^{n}P_r=\frac{n!}{\left(n-r\right)!}$

${}^{n}C_r=\frac{n!}{\left(n-r\right)!\times r!}$

Equating both as equal

$$\begin{gathered} {}^{n}P_r={}^{n}C_r \\ \frac{n!}{\left(n-r\right)!}=\frac{n!}{\left(n-r\right)!\times r!} \\ 1=\frac{1}{r!} \\ r!=1=1! \end{gathered}$$

Equating both side we get,

$r=1$

Hence, the value of $r$ is $1$ when ${}^{n}P_r$ and ${}^{n}C_r$ are equal.