Hello, azikamkar!
In poker. a Full House is a 5-card hand with 3 cards of one rank and 2 cards of another rank.
Suppose you are dealt 5 cards from a standard 52- card deck.
What is the probability of drawing any full house?
There are \(\displaystyle \begin{pmatrix}52\\5\end{pmatrix}\:=\;2,598,960\) possible poker hands.
To get a Full House, there are
13 choices for the rank of the 3-of-a-kind
. . and \(\displaystyle \begin{pmatrix}4\\3\end{pmatrix} =\)
4 ways to select the triple.
Then there are
12 choices for the rank of the pair
. . and \(\displaystyle \begin{pmatrix}4\\2\end{pmatrix} =\)
6 ways to select the pair.
Hence, there are:
.\(\displaystyle 13\,\times\,4\times\,12\,\times\,6\:=\:3744\) possible Full Houses.
Therefore:
.\(\displaystyle P(\text{Full House})\:=\:\frac{3744}{2.598,960} \:= \:\frac{6}{4165}\)
