$Probability of making a full house in a poker hand$

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$Probability of making a full house in a poker hand$

5 cards are drown at random from an ordinary deck of 52 cards. What is the probability of getting a full house?

I know the answer is 0.001441 or 0.1441 %, but I like to see how it is computed.

Combinatorics Discrete Mathematics

Mason Miller

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We need to have a three of a kind and a pair. We could have a three of a kind in
\[{13 \choose 1} {4 \choose 3}=13\times 4=52 \text{different ways}.\]
Next we want to get a pair from the remaining 12 kinds of cards
\[{12 \choose 1}{4 \choose 2}=12\times 6 =72 \text{different ways}.\]
So we can get a full hourse in
\[52 \times 72 \text{different ways}.\]
The total number of ways one can choose 5 cards out of 52 cards is
\[{52 \choose 5}=2598960.\]
So the probability of a full house is
\[\frac{52 \times 72}{2598960}= 0.00144057623.\]

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$Probability of making a full house in a poker hand$

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