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**