combination help

Question

 From a deck of 28 cards: the 8, 9, ... , Q, K, A of each of the four suits. How many 6-card hands are possible which satisfy:
the hand has exactly two hearts
the hand has exactly three spades

Answer 1

Kim,

 

Lets do the hearts....there are 7 hearts and we want to choose 2 in no particular order this can be done in ₇C₂ = 21 ways. There are ₂₁C₄=5985 ways to choose the remaining 4 cards. So the number of different hands is 21*5985=125,685.

The problem with 3 spades is done in the same way...you start out with 7 spades and 21 other cards, calculate how many ways you can choose 3 out of 7 and then how many ways you can choose 3 out of the remaining 21. I got 46,550

(₇C₃*₂₁C₃)

 

Jim