How many hands of 5 cards contain no hearts, and at least one heart?

Question

Standard deck of 52 cards.

Answer 1

There are 39 cards that are not hearts in a standard pack, so for the first card we have a choice of 39 cards. For the second and subsequent cards, there are 38, 37, 36 and 35 available in sequence. So that gives 39*38*37*36*35 divided  by the number of ways of arranging 5 objects=120. Therefore the number of hands is 39C5 or 39*...*35/120=575,757. 

There are 52C5=2,598,960 different hands possible from the whole pack. Of these there are 575,757 hands with no hearts. Therefore there are 2,598,960-575,757=2,023,203 hands containing at least one heart.