10. A standard deck of 52 cards has the usual 4 suits and 13 denominations. What is the probability that two cards selected at random, and without replacement, from this deck will have the same denomination or have the same suit?
Answer: $\frac{5}{17}$
Solution: Let $A$ be the event that the 2 chosen cards will have the same denomination; and let $B$ be the event that the 2 chosen cards will have the same suit. Note that $A \cap B=\emptyset$. So that $\mathbb{P}(A \cup B)=\mathbb{P}(A)+\mathbb{P}(B)$.

Since there are 4 suits to choose from, then there are ${ }_{4} C_{2}=6$ possible pairs of the same value. There are a total of 13 card values, so that
$$
\mathbb{P}(A)=\frac{(13)(6)}{{ }_{52} C_{2}}=\frac{1}{17}
$$

Since there are 13 card denominations to choose from, then there are ${ }_{13} C_{2}=78$ possible pairs of the same suit. There are a total of 4 suits, so that
$$
\mathbb{P}(B)=\frac{(4)(78)}{{ }_{52} C_{2}}=\frac{4}{17} .
$$

Thus, $\mathbb{P}(A \cup B)=\frac{1+4}{17}=\frac{5}{17}$.