This is Problem 1.58 from Tsitsiklis, Bertsekas, Introduction to Probability, 2nd edition.
We draw the top 7 cards from a well-shuffled standard 52-card deck.
Find the probability that: a) The 7 cards include exactly 3 aces. b)
The 7 cards include exactly 2 kings. c) The probability that the 7
cards include exactly 3 aces, or exactly 2 kings, or both
My solution was
a) $${3! {49 \choose 4} \over {52 \choose 7} }$$
b) $${2! {50 \choose 5} \over {52 \choose 7} }$$
c) $${3! {49 \choose 4} \over {52 \choose 7} } + {2! {50 \choose 5} \over {52 \choose 7} } – {3! 2! {47 \choose 2} \over {52 \choose 7} }$$
The solution book says
Why do they ${4 \choose 3}$ instead of (3!)?
We have 7 slots where we can put the cards with each slot accepting 1 card. We want to fill the first 3 with aces and the last 4 with non-aces. There are 3! ways to distribute aces into the first 3 slots.
And then there are ${49 \choose 4}$ ways to fill the remaining slots.
There is another card problem where they use that same logic to solve the problem: question, answer
Best Answer
a) Divide all cards into two groups: a group of $4$ aces and a group of $48$ non-ace cards. From the first group you select any $3$ aces and from the second group you select any $4$ cards.
b) Again divide all cards into two groups: a group of $4$ kings and a group of $48$ non-king cards. From the first group you select any $2$ kings and from the second group you select any $5$ cards.
c) Use inclusion-exclusion principle. Adding a) and b) will double count the outcomes of exactly 3 aces and 2 kings, which must be subtracted once. To find the outcomes of exactly 3 aces and 2 kings, you divide all cards into $3$ groups: a group of $4$ aces, a group of $4$ kings and a group of $44$ non-ace and non-king cards. From the first group you select any $3$ aces, from the second you select any $2$ kings and from the last group you select any $2$ cards.