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The problem you're asking about can be summarized as follows:

In a box, there are:
- 312 blue balls,
- 241 green balls,
- 132 yellow balls, and
- 211 red balls.

You want to determine the minimum number of balls you need to draw (without looking) to be sure that you will have at least:
- 10 blue balls,
- 20 green balls, and
- 100 yellow balls.

### Solution Approach:
This is a classical application of the "Pigeonhole Principle." To ensure you meet all the conditions, the worst-case scenario would involve drawing as many balls of the "wrong" colors as possible before reaching the required number of blue, green, and yellow balls.

1. **Blue Balls (10 required)**:
   To minimize the blue balls, you could first draw all the green, yellow, and red balls.
   - Green: 241
   - Yellow: 132
   - Red: 211
   This totals: \( 241 + 132 + 211 = 584 \)

   After these 584 balls, the next 10 balls must be blue to guarantee that you have 10 blue balls.
   So, in the worst case, you need to draw \( 584 + 10 = 594 \) balls to guarantee 10 blue balls.

2. **Green Balls (20 required)**:
   To minimize the green balls, you could first draw all the blue, yellow, and red balls.
   - Blue: 312
   - Yellow: 132
   - Red: 211
   This totals: \( 312 + 132 + 211 = 655 \)

   After these 655 balls, the next 20 balls must be green to guarantee that you have 20 green balls.
   So, in the worst case, you need to draw \( 655 + 20 = 675 \) balls to guarantee 20 green balls.

3. **Yellow Balls (100 required)**:
   To minimize the yellow balls, you could first draw all the blue, green, and red balls.
   - Blue: 312
   - Green: 241
   - Red: 211
   This totals: \( 312 + 241 + 211 = 764 \)

   After these 764 balls, the next 100 balls must be yellow to guarantee that you have 100 yellow balls.
   So, in the worst case, you need to draw \( 764 + 100 = 864 \) balls to guarantee 100 yellow balls.

### Final Answer:
The worst-case scenario is the largest of these three values: 864 balls. Therefore, you need to draw at least **864 balls** to be sure you will meet all the conditions.

Would you like more details or have further questions?

Here are 5 questions related to this problem:
1. How does the Pigeonhole Principle apply in this problem?
2. What if the problem asked for 50 blue balls instead of 10? How would the solution change?
3. Can this problem be solved with a different method besides the worst-case scenario approach?
4. What is the significance of calculating the "minimum number of draws" in probability problems like this?
5. Could we determine the probability of getting 10 blue balls before drawing 864?

**Tip:** When solving problems involving worst-case scenarios, always consider drawing the maximum number of other categories before reaching your goal category.

In a box, there are 312 blue, 241 green, 132 yellow, and 211 red balls. What is the minimum number of balls you need to draw without looking to ensure that you will have at least 10 blue balls, 20 green balls, and 100 yellow balls?

Solve a combinatorics problem to find the minimum number of balls drawn to guarantee at least 10 blue balls, 20 green balls, and 100 yellow balls. Learn about the Pigeonhole Principle and worst-case scenarios. Suitable for Grades 9-11.

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