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 3 pairs of slippers of 3 different colors and simultaneously 3 different styles?

(S. B. Gashkov)

## Answer: 21.
Solution. If you take no more than 20 slippers, it may happen that among them there will be no more than $20-12=8$ pairs, and they can quite possibly be not 3, but only 2 colors. Therefore, such a number of slippers is not enough to guarantee the desired 3 pairs of slippers.

If, however, you take 21 slippers, then among them there will be no less than $21-12=9$ pairs, and they will represent no less than 3 colors (otherwise, there would be no more than $3 \times 4=8$ pairs in total). But then we can choose 3 pairs of 3 different colors:
(a) if they are exactly 3 different styles, then these 3 pairs are the desired ones;

(b) if they are all of the same style, then we take any pair of a different style and replace the selected pair of the same color with it, ending up in the following situation (c);

(c) if they are exactly 2 different styles, then some 2 of the selected pairs have 1 style and at least 1 of them can be replaced by a pair of the same color but of some 3rd style (such a pair will be found, otherwise there would be no more than $2+2+4=8$ pairs in total), resulting in the desired 3 pairs.