Ensuring a Minimum Number of Balls of the Same Color in a Random Selection

Suppose you have a box containing 6 red, 8 green, 10 blue, 12 yellow, and 15 white balls. Your task is to determine the minimum number of balls you need to randomly select to ensure that you have at least 9 of the same color. This problem can be solved using the Pigeonhole Principle, which is a fundamental concept in combinatorics and number theory.

Understanding the Problem with the Pigeonhole Principle

The Pigeonhole Principle states that if ( n ) items are put into ( m ) containers and ( n > m ), then at least one container must contain more than one item. In our case, we want to ensure that we have at least 9 balls of the same color.

Step-by-Step Solution Using the Pigeonhole Principle

Let's break down the problem using the given quantities of balls in different colors:

Red: 6 Green: 8 Blue: 10 Yellow: 12 White: 15

Step 1: Identify the Maximum Number of Balls Without Reaching 9 of Any Color

We aim to determine the maximum number of balls we can take without having 9 of any color:

For red and green balls (which have 9 or fewer balls each):
6 red 8 green 14 balls. For blue, yellow, and white balls (which have more than 9 balls each):
Each of these colors can contribute up to 8 balls without reaching 9:
8 blue 8 yellow 8 white 24 balls.

Therefore, the maximum number of balls we can pick without having 9 of any color is:

14 (from red and green) 8 (from blue) 8 (from yellow) 8 (from white) 38 balls.

Step 2: Calculate the Minimum Number of Balls to Ensure 9 of the Same Color

To guarantee that we have at least 9 balls of one color, we need to take one more ball than the maximum calculated. Thus, the minimum number of balls we need to choose is:

38 1 39.

Conclusion

The minimum number of balls you need to randomly select to ensure that you have at least 9 balls of the same color is 39.

Remember, in a random selection, you might end up with 9 balls of the same color well before picking 39 balls. This is because if you pick the maximum number of balls from each color without reaching 9 (38 in total), the next ball you pick will inevitably be the 9th ball of one of the colors that have 8 balls already.

Equipped with the Pigeonhole Principle, you can solve similar problems and ensure that you meet the required criteria with a systematic approach.