Answer to POW #7:
There is a 12/22 chance that the first shoe George chooses is black.
Given this choice, 11 of the remaining 21 shoes are black, and 6 of
these are black AND belong on the opposite foot. Thus there is a probability
of (12/22)(6/21) that George will choose a matching pair of black shoes.
Similarly, there is a (6/22)(3/21) chance he will choose a matching pair of
brown shoes, and a (4/22)(2/21) chance he will choose a matching pair of
gray shoes.
Therefore the chance that George chooses a pair of shoes he can
wear and not look really silly is (12/22)(6/21) + (6/22)(3/21) + (4/22)(2/21)
= 49/231 = 7/33.
Source: MATHCOUNTS Handbook