Let $X$ be a random variable following the distribution $\mathcal{H}(n, p, A)$, i.e.
$$\mathbb{P}(X = k) = \frac{\binom{pA}{k}\binom{qA}{n-k}}{\binom{A}{n}} \quad \text{for all } k \in \llbracket 0, n \rrbracket.$$
We fix $n$ and $p$. Let $k \in \llbracket 0, n \rrbracket$. Show that
$$\lim_{A \to +\infty} \mathbb{P}(X = k) = \binom{n}{k} p^k (1-p)^{n-k}.$$