10. (50 points) In the kindergarten's changing room, in the lost and found basket, there are 30 mittens, of which 10 are blue, 10 are green, 10 are red, 15 are right-handed, and 15 are left-handed. Is it always possible to form sets of right and left mittens of the same color for 6 children?
Answer: No, not always.

Solution. There could have been 10 right red mittens, 10 left green mittens, and 5 right and 5 left blue mittens. From these, only 5 pairs of the same color (specifically, blue) can be formed.