Since the GMAT would use smaller numbers, let's change the problem as follows:
In a shop there are 6 different pairs of shoes. A man picks 4 shoes randomly. What is the probability that at least 1 pair of shoes is picked?
P(at least 1 pair) = 1 - P(no pairs).
P(no pairs):
The 1st shoe selected can be any of the 12 shoes.
P(2nd shoe does not match the 1st shoe) = 10/11. (Of the 11 remaining shoes, 10 do not match the 1st.)
P(3rd shoe does not match either of the first 2 shoes) = 8/10. (Of the 10 remaining shoes, 8 do not match either of the first 2 selected.)
P(4th shoe does not match any of the first 3 shoes) = 6/9. (Of the 9 remaining shoes, 6 do not match any of the first 3 selected.)
To combine these probabilities, we multiply:
10/11 * 8/10 * 6/9 = 16/33.
Thus:
P(at least 1 pair is selected) = 1 - 16/33 = 17/33.
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