How to Calculate the Number of Ways That Two Pairs May Be Dr

[Guest]

I am doing something wrong, when trying to calculate the number of hands that can be drawn, when dealing two pairs in a poker game, but can't see what it is, so I would appreciate being shown where my mistake is and--most importantly--WHY I am wrong, because I just can't see anything wrong with the way that I am doing it. My erroneous answer is exactly double what it should be, so I must be counting things twice.

Here is what I have (incorrectly) tried:

a. For the first pair, I choose one rank/denomination, then two suites: (13C1)(4C2)
b. Then, for the second pair, I again choose one rank/denomination from the remaining 12, then two suites: (12C1)(4C2)
c. Then, for the fifth card, I choose one rank/denomination from any of the remaining ranks/denominations, then one suite: (11C1)(4C1)
d. Finally, turning the crank, I get [(13C1)(4C2)][(12C1)(4C2)][(11C1)(4C1)] = (13)(6)(12)(6)(11)(4) = 247,104.

But the right way to do it is to choose both of the pairs' ranks/denominations simultaneously, like this:

[(13C2)(4C2)(4C2)][(11C1)(4C1)] = (78)(6)(6)(11)(4), which = 123,552.

Why do I have to choose both ranks/denominations at the same time, to start? What is wrong with my way of doing it?

[Guest]

Lets go through your procedure and generate a hand with 2 pairs:

a. For the first pair, I choose one rank/denomination, then two suites:
I'll choose rank 3, and for the suites clubs, hearts [So far my hand is {3 clubs 3 hearts}]
Because you have a 4C2 term, the order of my suites doesn't matter (choosing hearts clubs is the same as clubs hearts)

b. Then, for the second pair, I again choose one rank/denomination from the remaining 12, then two suites:
I'll choose rank 7, and for the suites clubs, spades [So far my hand is {3 clubs 3 hearts 7 clubs 7 spades}]

c. Then, for the fifth card, I choose one rank/denomination from any of the remaining ranks/denominations, then one suite:
I'll choose rank 8, and spades.

My hand is 3 clubs 3 hearts 7 clubs 7 spades 8 spades



Lets repeat with some different choices to get a different hand:

a. For the first pair, I choose one rank/denomination, then two suites:
I'll choose rank 7, and for the suites clubs, spades [So far my hand is {7 clubs 7 spades}]
My choice of rank was different this time so I should get a different hand.

b. Then, for the second pair, I again choose one rank/denomination from the remaining 12, then two suites:
I'll choose rank 3, and for the suites clubs, hearts [So far my hand is {7 clubs 7 spades 3 clubs 3 hearts}]

c. Then, for the fifth card, I choose one rank/denomination from any of the remaining ranks/denominations, then one suite:
I'll choose rank 8, and spades.

My hand is 7 clubs 7 spades 3 clubs 3 hearts 8 spades.

Oops, this is the same hand as before but with the pairs swapped. You forgot that the order of the rank of the pairs don't matter, so you are off by a factor of 2.