We are given the following information about sets A and B:
- A and B are non-singleton sets. This means the number of elements in A is greater than 1, and the number of elements in B is greater than 1. Mathematically, and .
- The number of elements in the Cartesian product of A and B is 35: .
- B is a proper subset of A: . This implies that all elements of B are also in A, and A contains at least one element not in B. Consequently, the number of elements in B must be strictly less than the number of elements in A: .
We need to find the value of the combination .
Step 1: Find n(A) and n(B).
We know that . So, we have:
We need to find pairs of integers ( ) such that their product is 35, and both integers are greater than 1 (from condition 1). The factors of 35 are 1, 5, 7, 35.
The possible pairs of factors greater than 1 whose product is 35 are (5, 7) and (7, 5).
Now, we use condition 3: .
- Case 1: If and , then . This contradicts condition 3.
- Case 2: If and , then . This satisfies condition 3. Both and satisfy condition 1.
Therefore, the only possibility is and .
Step 2: Calculate .
We need to calculate .
Using the formula for combinations:
Cancel out :
Thus, .
Comparing this result with the given options:
The correct option is 21.