There 52 balls in a box: 16 red, 16, blue, 16 yellow, and 4 that are either black or white. What is the least number of balls you must take out of the box (Without looking) to be sure that amount them there are 8 balls of the same color?
There 52 balls in a box: 16 red, 16, blue, 16 yellow, and 4 that are either black or white. What is the least number of balls you must take out of the box (Without looking) to be sure that amount them there are 8 balls of the same color?
The Correct Answer and Explanation is:
To be sure that you have 8 balls of the same color, the least number of balls you must take out of the box without looking is 37.
Explanation
This is a classic pigeonhole principle problem in probability and combinatorics. We are given a total of 52 balls of five possible colors:
- 16 red
- 16 blue
- 16 yellow
- 4 black or white (unspecified distribution, but worst-case scenario must be considered)
Our goal is to determine the minimum number of balls we must blindly pick to guarantee that at least 8 balls are of the same color, no matter the actual outcome of our picks.
Step 1: Consider the worst-case scenario
To delay reaching 8 of any single color, we can try to maximize variety in our picks.
Let’s assume the worst-case scenario:
- You pick 7 red, 7 blue, 7 yellow, 4 black, and 4 white balls.
- That’s a total of:
7+7+7+4+4=297 + 7 + 7 + 4 + 4 = 29 balls, and still no 8 balls of any color.
But wait! There’s a problem: the total number of black and white balls is only 4, not 4 each. So, we can’t have 4 black and 4 white in the worst case. The worst-case is if those 4 remaining balls are evenly split: 2 black and 2 white.
Step 2: Adjust for actual distribution
So, the worst-case color distribution you could pull without getting 8 of one color is:
- 7 red
- 7 blue
- 7 yellow
- 2 black
- 2 white
That totals:
7+7+7+2+2=257 + 7 + 7 + 2 + 2 = 25 balls
After 25 picks, it’s still possible that you don’t have 8 of any color.
Step 3: Next pick ensures success
On your 26th pick, no matter what color you draw, one of the first three main colors (red, blue, yellow) must reach 8, because you’ve already drawn 7 of each.
✅ Final Answer: 26 balls must be picked to guarantee 8 of the same color.
