The problem presented can be solved using the Pigeonhole Principle. Here's a breakdown of how to approach it:

Problem:

You have a box containing balls of different colors:

• 10 blue balls
• 20 red balls
• 8 green balls
• 15 yellow balls
• 25 white balls

You want to determine the minimum number of balls to choose to ensure that you have 12 balls of the same color.

Solution Using the Pigeonhole Principle:

The Pigeonhole Principle states that if $n$ items are distributed among $k$ containers, at least one container must contain more than $\left\lfloor \frac{n}{k} \right\rfloor$ items if $n > k$.

In this problem:

• The "containers" are the different colors of the balls (blue, red, green, yellow, and white), which gives us 5 different colors.
• We want to ensure that one of the colors reaches 12 balls.

Step-by-step reasoning:

1. Worst-case scenario: To avoid having 12 balls of the same color for as long as possible, we can assume we pick the maximum number of balls of each color without reaching 12 in any one color.

   • We can pick 11 balls of each color (since 11 is the maximum we can take from any color without reaching 12).
   • There are 5 colors, so the total number of balls we can pick without having 12 of the same color is: $11 \times 5 = 55 \text{ balls}$

2. Next ball ensures 12: As soon as we pick the 56th ball, we are forced to have 12 balls of at least one color, because we've already picked 11 of each color. The 56th ball must belong to one of these colors, ensuring that one color reaches 12 balls.

Thus, the minimum number of balls you must choose to guarantee that you have 12 balls of the same color is:

$\boxed{56}$

Would you like further details on how the Pigeonhole Principle works or any other clarification?

Related Questions:

1. How would the answer change if you wanted 13 balls of the same color?
2. What if there were 6 colors instead of 5? How would the calculation change?
3. How does the Pigeonhole Principle apply to other combinatorial problems?
4. Can you use the Pigeonhole Principle for non-color problems, like distributing people into groups?
5. What is the mathematical formula behind the Pigeonhole Principle in this context?

Tip:

When solving combinatorics problems, think in terms of worst-case scenarios. This helps ensure you're accounting for all possibilities!