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The question asks for the number of subsets \( S \) of the set \( \{1, 2, ..., 10\} \) such that \( \{1, 2, 3\} \cap S \) contains exactly one element.

To break this down:
1. You want \( S \cap \{1, 2, 3\} \) to contain exactly one element, meaning:
   - One of the elements from \( \{1, 2, 3\} \) must be in \( S \).
   - The other two elements from \( \{1, 2, 3\} \) must *not* be in \( S \).

2. The remaining elements, \( \{4, 5, 6, 7, 8, 9, 10\} \), can either be in \( S \) or not, and for each of these 7 elements, there are 2 choices (either in or out of \( S \)).

### Solution:
1. There are 3 ways to choose exactly one element from \( \{1, 2, 3\} \) (since there are 3 elements in total).
2. For each of the remaining 7 elements (\( \{4, 5, 6, 7, 8, 9, 10\} \)), there are 2 possibilities (in or out of \( S \)).

Thus, the total number of subsets \( S \) such that \( \{1, 2, 3\} \cap S \) has exactly one element is:
\[
3 \times 2^7 = 3 \times 128 = 384
\]

So, the correct answer is **384**.

Let me know if you'd like further details or have any questions!

### Related Questions:
1. What if we required \( S \cap \{1, 2, 3\} \) to have exactly two elements?
2. How many subsets of \( \{1, 2, ..., 10\} \) contain no elements from \( \{1, 2, 3\} \)?
3. How many subsets of \( \{1, 2, ..., 10\} \) contain all elements from \( \{1, 2, 3\} \)?
4. How does the number of subsets change if we increase the set to \( \{1, 2, ..., 12\} \)?
5. What is the total number of subsets of \( \{1, 2, ..., 10\} \)?

### Tip:
Remember, the number of subsets of a set with \( n \) elements is \( 2^n \), since each element can either be included or excluded from the subset.

Let S be a subset of {1, 2, ..., 10}. Find the number of S such that {1, 2, 3} ∩ S has exactly one element.

This problem involves counting subsets of the set {1, 2, ..., 10} such that the intersection with {1, 2, 3} has exactly one element. Learn how to use basic counting principles and combinatorics to solve this problem. Suitable for students in Grades 9-12.

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