Hypergeometric Mean and Variance via Indicator Variables

An urn contains 20 balls: 8 red and 12 blue. You draw 5 balls without replacement. Let $X$ be the number of red balls drawn. Define indicator variables $X_i = \mathbf{1}\{\text{ball } i \text{ is red}\}$ for each draw $i = 1, \ldots, 5$.

(a) Use linearity of expectation to find $E[X]$.

(b) Compute $\text{Cov}(X_i, X_j)$ for $i \ne j$ and use it to derive $\text{Var}(X)$.

(c) Verify that your variance formula reduces to the binomial variance $np(1-p)$ when $N \to \infty$ with $K/N \to p$ held fixed.