Sorry for the title, couldn't think of a better way to phrase it. The problem is this:
Forty children go to a carnival. Twenty-five are given a blue balloon, 30 a red balloon, 35 a green, and 33 a yellow. Prove that at least three children have all four colors of balloons.
I think I may have actually solved this one, but I'm not confident that my reasoning is proper. The way I view it: there are 123 total balloons distributed among the 40 children. If each child was given three of the four colors, there would be three balloons remaining. Therefore, at least three children must have received all four colors when the remaining three balloons are distributed.
Now, the issues I have with this solution are that it doesn't seem to guarantee the numbers given in the problem. What if only 24 were given a blue balloon, and 31 a red? My answer would be the same. Additionally, I don't see how my answer prevents any one child from receiving multiple of a balloon.
Could someone help explain to me the proper way to make my solution more rigorous? Thank you.
Your proof is correct! The only way to make it more rigorous would to start by assuming that at most 2 kids got all four colors, then go on to prove that there couldn't have been 123 balloons to start with.
Your proof also works for many other color distributions balloons, including the 24 blue, 31 red case you mentioned. The fact that no child can receive more than one balloon of the same color is implied by the problem wording, since it said "25 are given a blue balloon," not "25 blue balloons are given out."