I want to be good enough to do math competitions, and one of the main topics in them is always combinatorics. So I'm going through a combinatorics book, "A Walk through Combinatorics."
The confusion I have is with the following problem:
Find the prob that a poker hand has a pair( and nothing better than a pair).
My attempt: We know that a poker hand has five cards from a deck of 52 cards. Each card is equally likely so the total combinations is: $${52} \choose 5$$
The number of hands that contain only a pair and nothing better than a pair are: $${{13} \choose {4}} * {{4} \choose {2}}* {{4} \choose {1}}^3$$
There are 13 cards of each card. We choose 4 the 13 values. Then from one of the groups we want a pair, and from the remaining group we want a single.
So, the probability I got was:
$$\frac {{{13} \choose {4}} * {{4} \choose {2}}* {{4} \choose {1}}^3}{{52} \choose 5}$$.
However, according to the answers:
The correct probability should be : $\frac {{{13} \choose {1}} *{{12} \choose {3}} * {{4} \choose {2}}* {{4} \choose {1}}^3} {{52} \choose 5}$
Where is my logic flawed? Where am I going wrong? Am I double counting somewhere? I think my answer produces a smaller probability so am I undercounting?