Decks of Pinochle cards have a total of 48 cards and consist

Decks of Pinochle cards have a total of 48 cards and consist of 6 cards each of nines, tens, jacks, queens,
kings, and aces with there being two of each suit of each denomination (for example, there are 2 aces each of
diamonds, clubs, hearts, and spades for the total of 8 aces). Suppose that you are dealt a 12-card hand from a
deck of Pinochle cards. What is the probability that:
a) you are dealt exactly 4 hearts?
b) you are dealt at most 2 aces?

Solution

9) The 48 cards includes 12 spades (6 denominations * 2 = 12) and 36 non-spades

You have an 8-card hand. P(at most 2 spades) = P(0) + P(1) + P(2) =

C(12,0)C(36,8)/C(48,8) + C(12,1)C(36,7)/C(48,8) + C(12,2)C(36,6)/C(48,8) =

258986772/377348994 =1308014/1905803 =0.68633

b) As you have 2*4 = 8 kings, you have 40 non-kings.

Then, P(1) = C(8,1)*C(40,7)/C(48,8) = 149148480/377348994 = 24858080/62891499 = 0.3952