24. Provide the formula for the probability $P_{n}$ that among thirteen cards drawn from a full deck of 52 cards, $n$ cards will be of the spades suit.
Solution. In total, there are $C_{52}^{13}$ unordered sets that can be drawn. Among them, we are interested in sets where $n$ cards are of the spades suit (which can be chosen in $C_{13}^{n}$ ways) and $13-n$ cards are of another suit (which can be chosen in $C_{39}^{13-n}$ ways). The desired probability is

$$
P_{n}=\frac{C_{13}^{n} C_{39}^{13-n}}{C_{52}^{13}}
$$