In a drawer there are 6 pairs of size 42 black sneakers and 8 pairs of size 38 red sneakers. If they are drawn one by one and at random with the aim of obtaining a pair of the same color and size, in the worst case scenario, what is the number of slippers that must be removed to match a matching pair?
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Choices:
7
9
15
21
23
Other

To determine the number of slippers that must be removed in the worst-case scenario to obtain a matching pair of the same color and size, we need to consider the worst possible sequence of drawing slippers that delays finding a matching pair as long as possible.

Here's a step-by-step explanation:

1. **Total Slippers**: There are \(6 \times 2 = 12\) black slippers of size 42 and \(8 \times 2 = 16\) red slippers of size 38. This gives a total of \(12 + 16 = 28\) slippers.

2. **Worst Case Drawing**: In the worst case, we could draw one slipper of each size and color without getting a pair. 

3. **Possible Draws Without Pair**: 
   - If we draw all 12 black slippers (6 pairs of size 42) and all 16 red slippers (8 pairs of size 38), we would have drawn \(12 + 16 = 28\) slippers.
   - In the worst-case scenario, if we draw every slipper without drawing a matching pair, we will be left with one slipper for each size and color.

4. **Calculation**:
   - Since there are only two sizes and two colors, we can draw 21 slippers (i.e., 10 black size 42, 10 red size 38, and 1 additional slipper of any size/color) without getting a pair. 

5. **Worst Case for Drawing a Pair**:
   - Drawing 21 slippers ensures that each size and color has been exhausted to the point where the next draw will certainly result in a pair.

Therefore, in the worst-case scenario, you would need to draw **21** slippers to ensure you get at least one matching pair.

So, the correct answer is **21**.

Would you like more details or have any questions? Here are 5 additional questions you can ask:

1. How do we calculate the probability of drawing a matching pair of slippers in a given number of draws?
2. What are some combinatorial methods to solve problems involving drawing items randomly?
3. Can you explain how to calculate the expected number of draws to get a matching pair?
4. How would the problem change if there were an unequal number of left and right slippers?
5. What are the principles of the Pigeonhole theorem and how do they apply to this problem?

**Tip:** In combinatorics and probability, it's important to consider both the best and worst-case scenarios to understand the range of possible outcomes.

Explore the mathematical problem of finding matching pairs of sneakers from a drawer containing black and red sneakers of different sizes. Learn about probability and combinatorial methods used to determine the worst-case scenario for drawing matching pairs. Suitable for high school students studying probability and combinatorics.

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