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The question states:

> Ralph has 10 pairs each of white, red, and black gloves scattered in his workshop, all mixed together. In the dark, he began randomly picking individual glove pieces. What is the minimum number of gloves Ralph must pick, without looking, to ensure he gets at least one matching pair (for example, a left red glove and a right red glove)?

### Solution:
Ralph needs to ensure that he picks a **matching pair** of gloves (i.e., a left and a right glove of the same color). Here's how we can determine the minimum number:

1. **Worst-case scenario**: 
   If Ralph picks gloves randomly, he could pick all the left-hand gloves first (or all the right-hand gloves). There are **10 left-hand gloves** and **10 right-hand gloves** for each color (white, red, black).

   - Therefore, in the worst case, he might pick **all 30 gloves** (10 left gloves and 10 right gloves of each color), **none of which form a pair**.

2. **To ensure at least one pair**:
   If Ralph picks **one additional glove** (the 31st glove), this guarantees that he must form at least one matching pair because there are only 30 unmatched gloves total (10 left and 10 right of each color).

Thus, Ralph must pick **31 gloves** to ensure at least one matching pair.

### Correct Answer:
**(E) 31**

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Would you like a detailed explanation for any part of this? Here are five follow-up questions for deeper understanding:

1. How is the worst-case scenario calculated in similar problems?
2. What principle of combinatorics ensures this solution is valid?
3. How would the solution change if the number of pairs of gloves per color were different?
4. Can this problem be approached using a formal mathematical proof?
5. How does this logic apply to real-world problems of "ensuring pairs"?

**Tip:** Always consider the worst-case scenario in such problems to determine the guaranteed outcome.

Ralph has 10 pairs each of white, red, and black gloves scattered in his workshop, all mixed together. In the dark, he began randomly picking individual glove pieces. What is the minimum number of gloves Ralph must pick, without looking, to ensure he gets at least one matching pair (for example, a left red glove and a right red glove)?

Learn how to solve the problem of determining the minimum number of gloves Ralph must pick to ensure he gets a matching pair, using combinatorics and the pigeonhole principle. This problem involves understanding worst-case scenarios and guarantees. Suitable for students in Grades 6-8.

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