Five cards are drawn from a pack of 52 cards. What is the probability that exactly three of them are of the same suit.

(In reply to Puzzle Solution With Explanation by K Sengupta)

If we were required to deduce that exactly P cards belonged to the same suit, given that P are drawn from a pack of 52 cards, then proceeding similarly as before, the methodology would
be as follows:

We observe that the N cards of one suit can be chosen in
comb(13,Q), so that the remaining (P-Q) cards can be chosen out of the remaining (52-13) = 39 cards in comb(39,P- Q) ways.

Since there are a total of four suits to begin with, it follows that
the total number of ways to choose the 5 cards so that precisely
three of them belong to the same suit is  equal to 4*comb(13,Q)*comb(39,P-Q)

Now, the total number of ways to choose the 5 cards without any restriction = comb(52, P)

Consequently, the required probability is equal to:
4*comb(13,Q)*comb(39,P-Q)/ comb(52, P)

Substituting (P, Q) = (5, 3), we arrive at the probability sought in the given problem.

Edited on March 28, 2008, 4:45 am

Posted by K Sengupta on 2008-03-28 04:44:04