Can't get the solution to a question on combinations.

From a standard deck of cards how many 5 card hands are possible consisting of

a. exactly 4 hearts

b. two cards of one kind and three of another(like a full house).

Is part a  13!/9!4!?

part b not sure.



You have part a almost correct.  13!/9!4! is the number of ways of choosing 4 cards from the 13 hearts in the deck. But you want a 5 card hand with exactly 4 hearts. Thus you can select the final card from the 39 that are not hearts. Hence the number of 5 card hands with exactly 4 hearts is

39 x  13!/9!4!

For part b you can select the first kind in 13 possible ways. From the 4 cards of that kind select 3, which you can do in  4!/3!1! ways. Thus far you have

13 x  4!/3!1! = 52 ways to select the first 3 cards.

Now choose one of the 12 kinds rremaining and select 2 of the 4 cards of that kind. That you can do in

12 x  4!/2!2! = 72 ways.

Hence there are

52 x 72 = 3744 5 card hands with two cards of one kind and three of another.

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