2. Clowns Plukha and Shmyaka have six pairs of valenki (traditional Russian felt boots) between them. Each pair of valenki is painted in a unique color, and the valenki in a pair are identical (they are not distinguished as left or right). In how many ways can both clowns be simultaneously wearing mismatched valenki? (16 points)
Solution. We can use valenki (traditional Russian felt boots) from two, three, or four pairs.

1) We choose two pairs of valenki $C_{6}^{2}=15$ ways. Each clown puts on one valenok from different pairs, choosing which one for which foot. This gives us $15 \cdot 2 \cdot 2=60$ ways.
2) We use three pairs of valenki. We choose one complete pair of valenki, and two other pairs from which we will take one valenok each. This can be done $6 \cdot C_{5}^{2}=60$ ways. Each clown puts on one valenok from the pair, choosing which foot to put it on. This can be done in 4 ways. Then each clown chooses one of the two different valenki in 2 ways. This gives us $60 \cdot 4 \cdot 2=480$ ways.
3) We use four pairs of valenki. We choose four pairs from which we will take one valenok each. This can be done $C_{6}^{4}=15$ ways. The four chosen valenki of different colors can be put on the four feet of the clowns in 4! ways. This gives us $15 \cdot 4! =360$ ways.

Finally, we have $60+480+360=900$.

## Answer: 900.