\[\begin{align*} E[X] &= \sum_{x=1}^n x \frac{\binom{K}{x} \binom{N-K}{n-x}}{\binom{N}{n}} \\ &= \sum_{x=1}^n \frac{K\binom{K-1}{x-1} \binom{N-K}{n-x}}{\binom{N}{n}} \\ &= \frac{nK}{N} \sum_{x=1}^n \frac{\binom{K-1}{x-1}\binom{N-K}{n-1-(x-1)}}{\binom{N-1}{n-1}} \\ & \text{Using Vandermonde's Identity:} \\ & \text{let: } l = x-1 \\ &= \frac{nK}{N} \sum_{l=0}^{n-1} \frac{\binom{K-1}{l}\binom{N-K}{n-1-l}}{\binom{N-1}{n-1}} \\ &= \frac{nK}{N} \end{align*}\]
\[\begin{align*} E[X^2] &= \sum_{x=1}^n x^2 \frac{\binom{K}{x} \binom{N-K}{n-x}}{\binom{N}{n}} \\ &= \sum_{x=1}^n \frac{\left[K(K-1)\binom{K-2}{x-2} + K\binom{K-1}{x-1}\right] \binom{N-K}{n-x}}{\binom{N}{n}} \\ &= \frac{K(K-1)}{\binom{N}{n}}\sum_{x=1}^n\binom{K-2}{x-2}\binom{N-K}{n-x} + \frac{K}{\binom{N}{n}}\sum_{x=1}^n \binom{K-1}{x-1} \binom{N-K}{n-x}\\ & \text{Using Vandermonde's Identity} \\ & \text{let: } l = x-2 \text{ and } m = x-1\\ &= \frac{K(K-1)}{\binom{N}{n}}\sum_{l=0}^{n-2}\binom{K-2}{l}\binom{N-K}{n-2-l} + \frac{K}{\binom{N}{n}}\sum_{m=0}^{n-1} \binom{K-1}{m} \binom{N-K}{n-m-1}\\ &= \frac{K(K-1)n(n-1)}{N(N-1)} + \frac{nK}{N} \\ \end{align*}\]
Then:
\[\begin{align*} Var(X) &= E[X^2] - (E[X])^2 \\ &= \frac{K(K-1)n(n-1)}{N(N-1)} + \frac{nK}{N} -\left(\frac{nK}{N} \right)^2 \\ &= \frac{K(K-1)n(n-1)}{N(N-1)} + \frac{nK}{N} -\frac{n^2K^2}{N^2} \\ &= \frac{nK}{N}\left[\frac{(K-1)(n-1)}{(N-1)} + 1 - \frac{nK}{N} \right] \\ &= \frac{nK}{N}\left[\frac{N(K-1)(n-1)}{N(N-1)} + \frac{N(N-1)}{N(N-1)} - \frac{nK(N-1)}{N(N-1)} \right] \\ &= \frac{nK}{N}\left[\frac{N(K-1)(n-1) + N(N-1) -nK(N-1)}{N(N-1)}\right] \\ &= \frac{nK}{N^2(N-1)}\left[N(K-1)(n-1) + N(N-1) -nK(N-1)\right] \\ &= \frac{nK}{N^2(N-1)}\left[N(Kn-n-K+1) + N^2 - N - nKN + nK\right] \\ &= \frac{nK}{N^2(N-1)}\left[NKn-Nn-NK+N + N^2 - N - nKN + nK\right] \\ &= \frac{nK}{N^2(N-1)}\left[-Nn-NK + N^2 + nK\right] \\ &= \frac{nK}{N^2(N-1)}\left[N(N-K) - n(N-K)\right] \\ &= \frac{nK}{N^2(N-1)}\left[(N-n)(N-K)\right] \\ &= \frac{nK(N-n)(N-K)}{N^2(N-1)} \end{align*}\]
\[\begin{align*} M_X(t) &= E[e^{tX}] \\ &= \sum_{k=0}^n e^{tk} \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}} \\ &= \frac{1}{\binom{N}{n}}\sum_{k=0}^n e^{tk} \binom{K}{k} \binom{N-K}{n-k} \\ &= \frac{1}{\binom{N}{n}}\sum_{k=0}^n \binom{K}{k} (e^t)^k\binom{N-K}{n-k} \\ & \text{Using the Vandermonde Generalization, and Falling Factorials of the binomial expressions:} \\ &= \frac{1}{\binom{N}{n}} \sum_{k=0}^n \frac{(-1)^k(-K)_k}{k!} (e^t)^k \binom{N-K}{n}\frac{(-n)_k(-1)^k}{(N-K-n+1)_k} \\ &= \frac{\binom{N-K}{n}}{\binom{N}{n}} \sum_{k=0}^n \frac{(e^t)^k}{k!} \frac{(-K)_k(-n)_k}{(N-K-n+1)_k}\\ &= \frac{\binom{N-K}{n}}{\binom{N}{n}} {_2F_1}(-K, -n; N-K-n+1; e^t) \end{align*}\]