A poker hand consisting of 4 cards is dealt from a standard deck of 52 cards. Find the probability that the hand contains exactly 3 face cards. Leave your answer as a reduced fraction.

Answer 1

You can select the four cards in 52 C 4 ways

Suppose your hand has 3 face cards (J Q K) and 1 other card

There are 3 x 4 = 12 face cards in the pack, so we can choose 3 in 12 C 3 ways

There are 52 - 12 = 40 non face cards so we can choose 1 in 40 C 1 ways

So by counting principle we can choose the 4 cards in : 12 C 3 X 40 C 1 ways

So required probability is [ 12 C 3 x 40 C 1] / [ 52 C 4 ]

= (220 x 40) / (270725) = 8800 / 270725 = 1760 / 54145 = 352 / 10829

This cant be simplified further as the only odd factor of 352 is 11, which doesnt divide into 10829