# Combination question

• Jun 19th 2013, 03:33 AM
downthesun01
Combination question
In a poker hand of 5 cards from an ordinary deck of 52 cards, find the probability of getting a full house. Note: a full house is a hand that consists of 3 of one rank and 2 of another.

My solution:

$\displaystyle \frac{13{{4}\choose{3}} * 12{{4}\choose{2}}}{{{52}\choose{5}}}$$\displaystyle +\frac{12{{4}\choose{3}} * 13{{4}\choose{2}}}{{{52}\choose{5}}} But, for some reason my answer is twice the answer given in the solution. • Jun 19th 2013, 05:46 AM Plato Re: Combination question Quote: Originally Posted by downthesun01 In a poker hand of 5 cards from an ordinary deck of 52 cards, find the probability of getting a full house. Note: a full house is a hand that consists of 3 of one rank and 2 of another.My solution: \displaystyle \frac{13{{4}\choose{3}} * 12{{4}\choose{2}}}{{{52}\choose{5}}}$$\displaystyle +\frac{12{{4}\choose{3}} * 13{{4}\choose{2}}}{{{52}\choose{5}}}$
But, for some reason my answer is twice the answer given in the solution.

It is because you did the same count twice. WHY?
• Jun 19th 2013, 10:23 PM
downthesun01
Re: Combination question
I realized my mistake while in the car this morning. I multiplied both sides by the permutation of the rank pairs {AAAA},{2222},... instead of multiplying by the combination of rank pairs. My answer should have been:

$\displaystyle {{12\choose{2}}*\left(\frac{{{4}\choose{3}} * {{4}\choose{2}}}{{{52}\choose{5}}}$$\displaystyle +\frac{{{4}\choose{3}} * {{4}\choose{2}}}{{{52}\choose{5}}})$