There are 4 red, 5 green, and 6 blue balls inside a box. If N number of balls are picked simultaneously, what is the smallest value of N that guarantees there will be at least two... There are 4 red, 5 green, and 6 blue balls inside a box. If N number of balls are picked simultaneously, what is the smallest value of N that guarantees there will be at least two balls of the same colour?
Understand the Problem
The question is asking for the minimum number of balls to be picked in order to ensure that at least two balls of the same colour are chosen from a box containing 4 red, 5 green, and 6 blue balls. This involves applying the Pigeonhole Principle.
Answer
$N = 4$
Answer for screen readers
The minimum number of balls that need to be picked to guarantee at least two balls of the same color is $N = 4$.
Steps to Solve
- Identify the total colors of balls
In this problem, we have three different colors of balls: red, green, and blue.
- Count the number of each color
- Red balls: 4
- Green balls: 5
- Blue balls: 6
- Determine total unique colors
Since we have 3 colors (red, green, blue), if we pick one ball of each color, we will have 3 balls without any repetitions.
- Apply the Pigeonhole Principle
According to the Pigeonhole Principle, if we have (k) different categories (in this case, colors) and we want to ensure at least one category has at least two items, we need to pick (k + 1) items.
- Calculate the minimum number of picks required
Given that we have 3 colors, to ensure at least two balls of the same color are picked, we must choose: $$ N = 3 + 1 = 4 $$
The minimum number of balls that need to be picked to guarantee at least two balls of the same color is $N = 4$.
More Information
This problem utilizes the Pigeonhole Principle, which is a fundamental concept in combinatorial mathematics. The principle asserts that if more items are distributed among fewer categories than there are items, at least one category must contain more than one item.
Tips
- Miscounting the number of colors: Ensure to identify all distinct colors before applying the Pigeonhole Principle.
- Incorrectly calculating (N): Remember that (N) must be the total number of colors plus one to guarantee a duplicate.
