what am I doing wrong on this combination/probability: 12 cards

[imtrying]

Guys, I got this problem. Got several ways of solving it. One of them is incorrect but I don't seem to know why:

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, there are 6 same-value pairs in the deck. Let's shuffle the deck, turns over 4 cards, What is the chance that we will find at least one pair of cards that have the same value?




8/33


62/165


17/33


103/165


25/33








Without spoiling it in this post, in a followup post, I'll give how I solve it and got the right answer and how I solve it in a different way, which gave me a wrong answer. I don't get why the different method doesn't lead to the same answer.

[imtrying]

Seeing 'at least', I tried to get a solution for four cards of totally four different values:

Pick 4 different values from 6 values: 6C4. For each pair of the same value, there are 2 different ways of picking a card. Therefore, total number of ways of picking 4 different values : 6C4 * 2^4.

Total number of ways of picking 4 cards: 12C4.

So the answer is 1 - 6C4 * 16 / 12C4 = 1 - 16/33 = 17/33.


Here's a different strategy, please kindly tell me what is wrong with this:

Pick a pair of cards of the same value: 6C1. For the other two cards, they can be anything, so 10C2. Therefore total number of ways of picking four cards with at least two cards of the same value: 6C1 * 10C2

The answer would be 6C1 * 10C2 / 12C4 = 18/33

What is wrong with this approach?!

TIA!

[imtrying]

Please close this thread, as I have re-posted the question into the Problem Solving subforum:

http://www.urch.com/forums/gmat-prob...tml#post670158 (what am I doing wrong on this combination/probability: 12 cards)

[bose]

Imo C
12c4-6c4(2)^4
12c4=total # Of Ways
6c4=select 4 #s From 6#s
Each # Has Two Suits So 2 Ways, I.e. 2^4 For 4 #s