34. In bridge, a standard 52 -card deck is dealt in the usual way to 4 players. By convention, each hand is assigned a number of "points" based on the formula
$$
4 \times(\# \mathrm{~A} ' \mathrm{~s})+3 \times(\# \mathrm{~K} \text { 's })+2 \times(\# \mathrm{Q} \text { 's })+1 \times(\# \mathrm{~J} \text { 's }) \text {. }
$$

Given that a particular hand has exactly 4 cards that are A, K, Q, or J, find the probability that its point value is 13 or higher.
Answer: $\frac{197}{1820}$
Solution: Obviously, we can ignore the cards lower than J. Simply enumerate the ways to get at least 13 points: AAAA (1), AAAK (16), AAAQ (16), AAAJ (16), AAKK (36), AAKQ (96), AKKK (16). The numbers in parentheses represent the number of ways to choose the suits, given the choices for the values. We see that there are a total of $1+16+16+16+36+96+16=197$ ways to get at least 13. There are a total of $\binom{16}{4}=1820$ possible ways to choose 4 cards from the 16 total A's, K's, Q's, and J's. Hence the answer is $197 / 1820$.