How many poker hands have two pairs?

Question

I'm trying to calculate how many poker hands called Two Pair, there are. Such a hand consists of one pair of one rank, another pair of another rank and one card of a third rank. A poker hand consists of 5 cards.

I have two methods that I thought would work equally well. Turns out only one of them yields the correct answer. I was wondering if anyone here knows why the second solution gives the wrong answer.

Solution 1 (Correct):

We choose 2 ranks out of 13, which can be done in $\binom{13}{2}$ ways.

For the first rank we choose 2 suits out of 4, which can be done in $\binom{4}{2}$ ways.

For the second rank we choose 2 suits out of 4, which can be done in $\binom{4}{2}$ ways.

The last card can be chosen in $44$ different ways.

So the total number of hands is $\binom{13}{2}\cdot \binom{4}{2}\cdot \binom{4}{2}\cdot 44=123,552$

Solution 2 (Incorrect):

We choose 3 ranks out of 13, which can be done in $\binom{13}{3}$ ways.

For the first rank we choose 2 suits out of 4, which can be done in $\binom{4}{2}$ ways.

For the second rank we choose 2 suits out of 4, which can be done in $\binom{4}{2}$ ways.

For the third rank we choose 1 suit out of 4, which can be done in $4$ ways.

So the total number of hands is $\binom{13}{3}\cdot \binom{4}{2}\cdot \binom{4}{2}\cdot 4=41,184$

This is just a third of the correct number of hands. Why the second solution is wrong unfortunately seems to elude me......

Answer 1

Assume your choose three ranks R1, R2, R3.

In the first solution: You choose (R1, R2) first $\binom{13}{2}$ then assign the suits. Lastly, you choose R3 from the rest (remaining 44). The reason why $\binom{13}{2}$ is chosen to avoid double count since order does not matter for this two ranks: as (R1,R1,R2,R2,R3) is the same as (R2,R2,R1,R1,R3). (writing R1 R1 meaning two suits rank 1, pardon my laziness). This is correct.

In the second solution:

You choose (R1,R2,R3) first then assign the suits. But using $\binom{13}{3}$ means the order of the three ranks do not matter. In other words, you are treating the three hands (R1,R1,R2,R2,R3), (R3,R3,R2,R2,R1), (R1,R1,R3,R3,R2) as one hand only. Thus you undercount 3 times.

Answer 2

After you've chosen which three ranks are in the hand, you need to choose either (a) which two of the three ranks to make the pairs, or (b) which one of the ranks to make the singleton. The number of ways to do these are ${3 \choose 2}$ and ${3 \choose 1}$, respectively, and of course each equals $3$.

Answer 3

Your first method is to count ways to choose two from thirteen ranks for the pairs, two from four suits for each of those, and one from fourty-four cards to be the loner (or one from eleven ranks and one from four suits).   That is okay. $$\binom{13}2\binom 42^2\binom{44}1 \\\binom{13}2\binom 42^2\binom {11}1\binom 41$$

Your second method is to count ways to choose three from thirteen ranks, two from four suits for the pairs, one from four suits for the singleton, and—wait—which two from those three selected ranks are to be the pairs?   Ah, that is better.$$\binom {13}3\binom 42^2\binom 41\binom 32$$

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...and of course $\binom{13}{3}\binom 32=\frac{13!}{3!10!}\frac{3!}{2!1!}=\frac{13!}{2!11!}\frac{11!}{10!1!}=\binom {13}2\binom{11}1$