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.