Problema Solution
If five cards are chosen at random from a standard deck of playing cards, how many different ways are there to draw the five cards if at least three cards are a jack, queen or a king?
Answer provided by our tutors
We will assume that the total number of cards is 52.
The number of different ways to draw three cards that are jack, queen or a king is:
C(12, 3)*C((52 - 12), 2) = (12!/(9!3!))*(40!/(38!2!) = (10*11*12/6)(39*40/2) = 171,600
The number of different ways to draw four cards that are jack, queen or a king is:
C(12, 4) C((52 - 12), 1) = (12!/(8!4!))*(40!/(39!1!) = (9*10*11*12/24)*40 = 19,800
The number of different ways to draw five cards that are jack, queen or a king is:
C(12, 5) = 12!/((12 - 5)!5!) = (8*9*10*11*12)/120 = 792
If we add them all we get the answer:
171,600 + 19,800 + 792 = 192,192 ways.