Prove the formula for the mean of the hypergeometric distribution with the parameters \(n, a\), and \(N\), namely, \(\mu=n \cdot \frac{a}{N}\).

\[\sum_{r=0}^{k}\left(\begin{array}{c}m \\end{array}\right)\left(\begin{array}{c}s \\k-r\end{array}\right)=\left(\begin{array}{c}m+s \\k\end{array}\right)\]

which can be obtained by equating the coefficients of \(x^{k}\) in \((1+x)^{m}(1+x)^{s}\) and in \((1+x)^{m+s}\).]