Sorry Alan
Here is another approach:
We will consider these by the cases of what our first shoe selection is 12/22 . If our first shoe is black, which happens with probability 6/21 , then our second shoe will be black and for the opposite foot with probability (6/22)(3/21) . Likewise, for brown shoes, our probability is the product (4/22)(2/21) . And for gray, . So the sum is equal to:
(12 * 6 + 6 * 3 + 4 * 2)/(22 * 21)
98 / (33 * 14)
7/33
+356 
