### Instruction: We have a standard deck of 52 cards, with 4 cards in each of 13 ranks.  We call a 5-card poker hand a full house if the hand has 3 cards of one rank and 2 cards of another rank (such as 33355 or AAAKK).  What is the probability that five cards chosen at random form a full house?### Response: I need to count the number of possible full house hands and divide it by the number of possible 5-card hands.
To form a full house, I need to choose a rank for the 3-of-a-kind, then choose 3 cards from that rank, then choose another rank for the pair, then choose 2 cards from that rank.
There are 13 ways to choose the first rank, and 4 choose 3 = 4 ways to choose 3 cards from that rank.
Then there are 12 ways to choose the second rank, since I can't repeat the first one, and 4 choose 2 = 6 ways to choose 2 cards from that rank.
So the number of possible full house hands is 13 times 4 times 12 times 6, which is 3744.
The number of possible 5-card hands is 52 choose 5, which is 2598960.
So the probability of getting a full house is 3744 divided by 2598960, which is 0.001440576.