Question

I have 500 red balls, 500 green balls, 500 blue balls, and 500 yellow balls in a bin. What is the smallest amount of balls I need to grab so that I am sure I have grabbed 250 balls of one color?

Answer 1

997

Explanation:

I multiplied 249 by 4 first. This is because I wanted to find how many balls the person could grab in total without having 250 of the same color.

249*4= 996

However, the question wants to guarantee that one color has 250.

So I did 996+1=997. (which means there could be: 249 red, 249 yellow, 249 green, and 250 blue)

This means that every other color could be at 249, but one would guarantee to be at 250 with 997 pulls

Answer 2

To ensure that you have grabbed 250 balls of one color, you need to use the Pigeonhole Principle. The worst-case scenario is that you grab 249 balls of each color, totaling 996 balls. Therefore, the smallest amount of balls you need to grab is 997, which guarantees that you have grabbed 250 balls of one color.

To explain further, let's consider the worst-case scenario where you grab 249 red balls, 249 green balls, 249 blue balls, and 249 yellow balls. This totals to 996 balls. However, if you grab just one more ball, you will have a total of 997 balls, and since there are only four colors, at least one color will have 250 balls or more. Therefore, by grabbing 997 balls, you can be sure that you have grabbed at least 250 balls of one color.