Answer
The value of $\frac{n!}{\left( n-r \right)!}$ for $ n=20$ and $ r=3$ is $6840$.
Work Step by Step
The given expression is $\frac{n!}{\left( n-r \right)!}$.
Put $ n=20$ and $ r=3$ in the expression $\frac{n!}{\left( n-r \right)!}$ and simplify,
$\begin{align}
& \frac{n!}{\left( n-r \right)!}=\frac{20!}{\left( 20-3 \right)!} \\
& =\frac{20!}{17!}
\end{align}$
It is know that
$\begin{align}
& n!=n\times \left( n-1 \right)\times \left( n-2 \right)\times \left( n-3 \right)..........3\times 2\times 1 \\
& =n\times \left( n-1 \right)!
\end{align}$
$\begin{align}
& n!=\frac{20\times 19\times 18\times 17!}{17!} \\
& =20\times 19\times 18 \\
& =6,840
\end{align}$
Thus, the value of $\frac{n!}{\left( n-r \right)!}$ for $ n=20$ and $ r=3$ is $6840$.
