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To determine which collections of sets form a partition of the set \( A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \), we need to check the following conditions for each collection:

1. **All elements of \( A \) must be included in the sets**.
2. **Each element of \( A \) should appear exactly once across the sets** (no overlap).
3. **There should be no empty sets**, except if the set contains just the empty set (which would not form a partition of \( A \)).

Now, I'll check each option:

### Option 1:
- Sets: \( \{2, 3, 5\}, \{1, 4, 7\}, \{8, 6, 9\} \)
- Missing element: \( 10 \)
- This is **not** a valid partition.

### Option 2:
- Sets: \( \{2, 4, 8, 10\}, \{1, 3, 7\}, \{5, 6, 9\} \)
- All elements from \( 1 \) to \( 10 \) are covered with no overlap.
- This is a **valid partition**.

### Option 3:
- Sets: \( \{4, 2, 6\}, \{10\}, \{8, 7, 1\}, \{5, 6, 3, 9\} \)
- Element \( 6 \) appears twice, so it's **not** a valid partition.

### Option 4:
- Sets: \( \{2, 9, 5, 11\}, \{1, 4, 3, 10\}, \{8, 7, 6\} \)
- Element \( 11 \) is not in the set \( A \), so it's **not** a valid partition.

### Option 5:
- Sets: \( \{1, 6\}, \{2, 7\}, \{3, 8\}, \{4, 9\}, \{5, 10\} \)
- All elements from \( 1 \) to \( 10 \) are covered with no overlap.
- This is a **valid partition**.

### Option 6:
- Sets: \( \{1, 6\}, \{2, 7\}, \{3, 8\}, \{4, 9\}, \{5, 10\}, \{\} \)
- There's an empty set, so it's **not** a valid partition.

### Option 7:
- Sets: \( \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)
- This is just the full set as a single subset, which **is** a valid partition.

### Final Answer:
The correct options are **2, 5, and 7**.

Would you like a deeper explanation on partitions or any other details? Here are 5 related questions:

1. What is the difference between a partition and a subset?
2. How are partitions applied in probability theory?
3. What is an example of a partition of a different set?
4. Can partitions be applied in solving combinatorial problems?
5. Why do empty sets not belong in valid partitions?

**Tip:** Partitions are fundamental in set theory, and they help in organizing elements into distinct, non-overlapping groups.

Which of the following collections of sets form a partition of the set A = {1,2,3,4,5,6,7,8,9,10}?

Explore how to determine which collections of sets form a partition of the set A = {1,2,3,4,5,6,7,8,9,10}. Understand the conditions for valid partitions in set theory and how to verify them.

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