I'm trying to calculate The probability of getting Four of a kind Aces with King high, and I don't know exactly how to do that.
My first way: To get for Aces I need to choose $4$ from $4$ to get King I need to choose $1$ from $4$ then divide all by choosing $5$ from $52$, so my equation is: $P = \frac{\binom{4}{4}\binom{4}{1}}{\binom{52}{5}}$
Now, I also was thinking to do it this way:Firs card has to be Ace so I need to choose $1$ from $4$, the second card has to be Ace as well and I used one Ace already so I need to choose $1$ from $3$, now repeat the same way for the rest tow cards, the last card has to be a King so need to choose $1$ from $4$ then divide all by choosing $5$ from $52$. The equation is: $P = \frac{\binom{4}{1}\binom{3}{1}\binom{2}{1}\binom{1}{1}\binom{4}{1}}{\binom{52}{5}}$
But both will give different results. Wich way is the correct way , and why the other way is incorrect?
The first calculation is correct. In the second calculation you’ve counted each of the four hands $24$ times, once for each of the $4!=24$ possible orders in which you could list the four aces. When you say that there are $4$ ways to choose the first ace, $3$ ways to choose the second, $2$ ways to choose the third, and one way to choose the fourth, you’re counting picking
picking the ace of spades first, the ace of clubs second, the ace of diamonds third, and the ace of hearts last, and
picking the ace of clubs first, the ace of hearts second, the ace of clubs third, and the ace of diamonds last
as giving you two different hands, when in fact they both give you the same four cards.
Notice that there really are very clearly only four of these hands: you have to have the four aces, so the only difference between two hands is which king they have. There are only four kings, so there are only four possible hands of this kind. Thus, you can go straight to
$$\frac4{\binom{52}5}\;.$$