A box contains 6 red, 8 green, 10 blue, 12 yellow and 15 white balls. What is the minimum number of balls to choose so we get 9 balls of the same color?
I know that the answer to this problem is 6 + 8 + 8 + 8 + 8 + 1 = 39. However, according to the generalized pigeonhole principle the answer should be the minimum integer that satisfies $\frac{N}{5} \geq 9$, which is 41. Why these to answers differ?
I think it's because there can't be 9 red or green balls, but I don't see exactly why 2 should be subtracted from 41 to get the correct answer.
If you had plenty balls of each color, then you could get 8 balls of each color, for a total of 40 balls, and so only the 41st would force there to be a color with 9 balls. However, you only have 6 red balls, so that accounts for the difference of 2.