Pigeonhole Principle Rainbow Problem

Question

There are 40 rainbows. Rainbows are only made out of up to 4 colors: Red, Orange, Blue, and Purple. 25 of the rainbows have red, 30 of them have orange, 33 of them have blue, and 35 of them have purple. Prove that there exists at least 3 rainbows such that each of the rainbows have all 4 of the possible colors.

I'm pretty sure this is just a basic application of the Pigeonhole Principle, but I'm not really sure how to get started. The holes I have right now are the 4 colors. Any help would be greatly appreciated.

Answer 1

Call each instance of a color in a rainbow a rainbow-color. The $40$ rainbows contain altogether $25+30+33+35=123$ rainbow-colors. If each rainbow contained only $3$ colors, that would account for only $120$ rainbow-colors. No rainbow can account for more than $4$ rainbow-colors. Can you finish it now?

Answer 2

$15$ of them do not have red. $10$ of them do not have orange. $7$ of them do not contain blues. And $5$ of them do not contain purple. So at the very most there are $37$ that ar lacking colors. But there are $40$ rainbows so at least $3$ of them that do not lack any colors.