Manhattan GMAT

This one made me think for atleast 5 min..

In any case here's my soln

Prob of having atleast one pair = 1 - prob of having NO matching pair

Prob of having NO matching pair = # of ways selected cards have no pair/# of ways of selection of 4 cards

# of ways of selection of 4 cards from 12 = 12C4 = 11*9*5

# of ways such that selected cards have NO pair = Sigma 6Ck * (6-k)C(4-k) where k ranges from 1 to 4

for eg for k=0 the eqn => we r selecting 0 cards from suit 1 and 4 cards from suit 2 so that no pair is formed

for k=1 => we r selecting 1 card from suit 1 and 3 cards from remaining 5 cards in suit 2 such that no pair is formed. Note that now we have to do selection from only 5 but not 6 cards in suit 2 coz we r making sure that the corresponding card for suit 1 is removed in suit 2.

and so on u can interpret for other k values

summing up we get # of ways such that selected cards have NO pair = 240

=> reqd prob = 1 - 240/( 11*9*50 = 1 - 16/33 = 17/33

PS: Looks like a cumbersome soln but this was the first soln i cud get. Wud appreciate if anyone can post a better approach.

Neo

Hi ,

We need to select 4 cards out of 12 cards.
letz go step by step.

We need to 1st find out probability of selecting zero similar cards and subtract it from 1 to get the answer.

1) To pick 1st card :
NO of possible Cards : 12.
we need to select any card.
ie Probability is 1.
Probability of selecting 1st card is 1.

2) To pick 2nd Card :
No of Possible Cards : 12 - 1 = 11 (one card is already selected)
No of cards we can choose which is not similar.
11 - 1 = 10 (We have already selected 1 card. Out of 11 cards there is one card which is similar to this selected card. We don't want to select it.)
Probability of selecting 2st card is 10/11

3) To Pick 3rd Card :
No of Possible Cards : 12 - 2 = 10 (2 cards are already selected)
No of cards we can choose which is not similar. 10 - 2 = 8
(We have already selected 2 card. Out of 10 cards there is two card which is similar to this selected card. We don't want to select it.)

Probability of selecting 3rd card is 8/10


4) To pick 4th Card :
No of Possible Cards : 12 - 3 = 9 (3 cards are already selected)
No of cards we can choose which is not similar. 9 - 3 = 6
(We have already selected 3 card. Out of 9 cards there is three card which is similar to this selected card. We don't want to select it.)
Probability of selecting 4th card is 6/9


Hence. Total Probability is :

1* 10/11 * 8/10 * 6/9 = 16/33

So required answer = 1 - 16/33
= 17/33


Hope this method help you.