Question

Let the number of elements in sets \(A\) and \(B\) be five and two respectively. Then the number of subsets of \(A \times B\) each having at least 3 and at most 6 elements is:

• A. 752
• B. 772
• C. 792
• D. 782

Hint:

Use Pascal's triangle to quickly calculate combinations for small values of \(n\) and \(r\).

Answer 1

The Correct Option is C

To solve this problem, we need to find the number of subsets of \(A \times B\) that have at least 3 elements and at most 6 elements. Let's proceed step-by-step:

Step 1: Determining the Number of Elements in \(A \times B\)

The number of elements in the Cartesian product \(A \times B\) is equal to the product of the number of elements in sets \(A\) and \(B\).

\(|A \times B| = |A| \times |B| = 5 \times 2 = 10\)

Step 2: Calculate Total Number of Subsets

The total number of subsets for a set with 10 elements is given by \(2^{10}\).

\(2^{10} = 1024\)

Step 3: Calculate Subsets with Specific Sizes

We need subsets that have at least 3 elements and at most 6 elements. This means subsets can have 3, 4, 5, or 6 elements.

Step 4: Using Combinatorics to Find the Number of Subsets

• \(\text{Subsets with 3 elements} = \binom{10}{3} = 120\)
• \(\text{Subsets with 4 elements} = \binom{10}{4} = 210\)
• \(\text{Subsets with 5 elements} = \binom{10}{5} = 252\)
• \(\text{Subsets with 6 elements} = \binom{10}{6} = 210\)

Step 5: Sum of Required Subsets

Add the number of subsets for each subset size from 3 to 6.

\(120 + 210 + 252 + 210 = 792\)

Conclusion

The number of subsets of \(A \times B\) each having at least 3 and at most 6 elements is 792.

Thus, the correct answer is 792.