Deck of cards

[GmatKiss]

Bill has a small deck of 12 playing cards made up of only 2 suits of 6 cards each. Each of the 6 cards within a suit has a different value from 1 to 6; thus, for each value from 1 to 6, there are two cards in the deck with that value. Bill likes to play a game in which he shuffles the deck, turns over 4 cards, and looks for pairs of cards that have the same value. What is the chance that Bill finds at least one pair of cards that have the same value?

8/33
62/165
17/33
103/165
25/33

[coolhabhi]

This problem can be solved easily if the opposite is considered, i.e., if no pair is selected.

the probability for no pair to be selected is : (12/12)(10/11)(8/10)(6/9)

(12/12) is the probability of selecting any card
(10/11) is the probability of selecting any card other than the pair number of the first card
(8/10) is the probability of selecting any card other than the previous card numbers
(6/9) is the probability of selecting any card other than the previous card numbers

So the probability for no pair to be selected is : (12/12)(10/11)(8/10)(6/9) = 16/33

So the probability for at least one pair to be selected is 1- (16/33) = 17/33

Answer is C.