Question

what is the probability of getting 3 of a kind in poker?

3 of a kind is:
Any three cards of the same rank with two other different cards.

I know the answer it suppose to be 54912/2598960

But how do you get that?

Answer 1

there are 13 ranks from which the 3 cards can come from, plus 2 more from the remaining
no of ways to get 3 cards = 13*C(4,3)
but we need two more cards from the remaining 48 cards or C(48,2)
so no of ways to choose 3 of a kind
= 13*C(4,3)*C(48,2) = 58656

but that would include the cases where the other two cards could be two-of-a-kind with a different rank.
We can't have that or our 3 of a kine would be a full house.
so we have to subtract the number of ways to get a full house which would be
13*C(4,3)*12*C(4,2) = 3744

so the prob of 3 of a kind
= (58656-3744)/C(52,5)
= 54912/2598960
= 3432/162435