2. In a dark room on the warehouse, 24 slippers are scattered, which originally formed 12 pairs: 3 different colors and 4 different styles (there were no identical pairs). What is the smallest number of slippers the seller must take out of the room to definitely present the customer with 2 pairs of slippers of different colors and simultaneously different styles?

(S. B. Gashkov)

Answer: 17.
Solution. If you take no more than 16 slippers, it may happen that among them there will be no more than $16-12=4$ pairs, and all of them could be of the same color. Therefore, such a number of slippers is not sufficient to guarantee the required 2 pairs of different colors and styles.

If, however, you take 17 slippers, then among them there will be no less than $17-12=5$ pairs, and they will represent no less than 2 colors (since there will be no more than 4 pairs of each color) and no less than 2 styles (since there will be no more than 3 pairs of each style). Then, we can choose 2 pairs of different colors from them:

(a) if they are also of different styles, then these 2 pairs are the ones we are looking for;
(b) if they are of the same style, then we add a pair of a different style, and it will be of a different color from at least one of the selected 2 pairs, thus forming the required 2 pairs.