### 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: To find the probability of a full house, I need to divide the number of ways to make a full house by the total number of possible 5-card hands.
The total number of possible 5-card hands is the number of ways to choose 5 cards from 52, which is 52 choose 5.
That's 52 factorial over 5 factorial times 47 factorial, which is the same as 52 times 51 times 50 times 49 times 48 over 5 times 4 times 3 times 2 times 1.
I can cancel some factors to simplify this.
I get 2,598,960.
Now, how many ways can I make a full house?
First, I need to choose the rank of the 3 cards, which can be any of the 13 ranks.
That's 13 choose 1, or just 13.
Then, I need to choose which 3 of the 4 cards of that rank I want.
That's 4 choose 3, or 4.
Next, I need to choose the rank of the 2 cards, which can be any of the remaining 12 ranks.
That's 12 choose 1, or 12.
Finally, I need to choose which 2 of the 4 cards of that rank I want.
That's 4 choose 2, or 6.
So, the number of ways to make a full house is 13 times 4 times 12 times 6, which is 3,744.