Probability of a poker hand

Question

Five cards are drawn from a shuffled deck of 52 cards without replacement.

What is the probability that the five cards will display a full house (three of a kind and two of a kind)?

Explain your answer.

Answer 1

Probability (FH) = Number of ways to get 3 of a kind * Number of ways to get 2 of a kind of a different card divided by the number of ways to get 5 cards from 52.

13 card values to get a TOK * 4 ways to pull 3 cards out of 4 * 12 card values to get 2 cards(can't be same a as TOK) * combination of 4 cards 2 at a time = 4*3/2 = 6

13*4*12*6/ ₅₂C₅ = 3744/2598960 = .00144 _nC_k = n!/(k!(n-k)!)

The formal way:

₁₃C_{1 4}C_{3 12}C_{1 4}C₂ / ₅₂C₅