Cartesian Product Of Sets

We are given the following information about sets A and B:

1. 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 .
2. The number of elements in the Cartesian product of A and B is 35: .
3. 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:

• 28
• 35
• 42
• 21

The correct option is 21.