Which of the following is equal to the binomial coefficient $\left(\genfrac{}{}{0ex}{}{n}{r}\right)$?

Solution

1. The definition is $\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\frac{n!}{r!(n-r)!}$.
2. The expression $\left(\genfrac{}{}{0ex}{}{n}{n-r}\right)$ is $\frac{n!}{(n-r)!(n-(n-r))!}$.
3. This simplifies to $\frac{n!}{(n-r)!r!}$, which is identical to $\left(\genfrac{}{}{0ex}{}{n}{r}\right)$.
4. Hence the answer is (A).