Picking two balls from 6 balls

Question

There are 30 balls in a bag. There are 6 balls each of 5 different colors. What is the minimum number of times I have to pick a ball to pick 2 balls of each color?

1. 24
2. 25
3. 26
4. 27
5. 28

Is the question correct? Someone asks me this and I'm confused.

Answer 1

If we want to have certainty of picking at least two balls of each colour. The worst case is if we have picked up all the balls of every colour but yellow, and only one yellow. So $25$ is not enough.

But $26$ is enough. For if we pick $26$, there are only $4$ balls left, so we have at least $2$ balls of each colour.

Remark: If on the other hand we consider the simpler problem of what is the least number of times we need to do the picking, if we are very lucky, then the answer is $10$, since with some luck when we pick $10$ balls we will have $2$ of each colour.

There is some (well, more than some) ambiguity in the question. The choices provided indicate that the problem-poser did not intend the question to be interpreted in the "if we get lucky" sense.

Answer 2

What we're counting is the minimum number of balls we could blindly pick yet be entirely sure we have two of each color. For example, liken it to picking socks blindly from a drawer -- we want to determine how many socks we have to pull out blindly to ensure we have a pair.

Think of the worst case scenario, where you go through all 6 of the first 4 colors before you get a ball of the last color. This means you would need to pick $4\times6$ balls before getting to your last color and to pick 2 of the last color we end with $4\times6+2=26$ balls.