You draw at random five cards from a standard deck of 52 cards. What is the probability that there is an ace among the five cards and a king or queen?
I want to do this by counting. There are 3 mutually exclusive events that need to be summed: ace and king with no queen, ace and queen with no king, ace and both.
Ace and king with no queen:
$$4 * 4 * {46 \choose 3} $$
One of 4 aces, one of 4 kings, 3 from the 46 other possible cards. Same thing for ace with queen and no kings.
Ace with both:
$$4 * 4 * 4 * {49 \choose 2} $$
One of 4 aces, one of 4 kings, one of 4 queens, 2 from the 49 cards remaining.
There are a total of $52 \choose 5$ ways to draw hands of 5 cards, so the answer is:
$$ \frac{2 * 4 * 4 * {46 \choose 3} + 4 * 4 * 4 * {49 \choose 2}}{52 \choose 5} = 0.215$$
...which is wrong. The answer is $0.1765$, which was obtained by taking $1 - P(A\cup B)$, where $A$ is no aces and $B$ is no king or queen. Where did I go wrong?