Probability of getting three of a kind in a Poker hand I have seen some answers to the question as
$$\frac{\displaystyle\binom{13}{1}\binom{4}{3}\binom{12}{2}\binom{4}{1}^2}{\displaystyle\binom{52}{5}} $$
Why don't we use
$$\frac{\displaystyle\binom{13}{1}\binom{4}{3}\binom{12}{1}\binom{11}{1}\binom{4}{1}^2}{\displaystyle\binom{52}{5}} $$
Which makes sense to me. There are 13 ranks and you choose 1. Of the one chosen there are 4 cards of which you choose 3. Now there are 12 ranks of which one is chosen. Now 4 choose 1. Leaving for the final card 11 ranks of which one is chosen. And choose 1 of 4. Divide all this by the number hands
The explanation that is given is that (aaabc)=(aaacb). But why? IF c came first, THEN call it b.
 A: The only difference between your calculation and the correct one is the selection of $\binom{12}{1}\times\binom{11}{1}$ versus $\binom{12}{2}$.
The first, your method, involves picking one suit, then picking another suit. The second involves picking both suits at the same time.
Your method double-counts the ways to select the other two, non-matched cards. If your three-of-a-kind aces go together with the king of spades and the queen of diamonds, your method will count the five cards altogether as two different hands. The other method counts it as just one hand, because it doesn matter when you got the aces and king and queen, just that you got them specifically.
This doesn't work because your method of counting all possible hands (in the denominator) must match. Instead it has checked for all combinations of five cards without considering the order, so that's the method you must use in the numerator as well.
A: Explaining the correct numerator.
$$13\times\binom{4}{3}\times\binom{12}{2}\times 4^2=54{.}912$$
This because

*

*you have 13 ways to choose the set


*$\binom{4}{3}=4$ ways to choose the suits of your set


*you have $\binom{12}{2}$ ways to choose the remaining (and different) cards where order does not matter.


*you have $4^2$ ways to choose the suits for the 2 remaining cards
