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

Step 1: Find the total number of elements in \(A \times B\).
- The Cartesian product \(A \times B\) contains \(n(A) \times n(B) = 5 \times 2 = 10\) elements. 
Step 2: Calculate the subsets.
- The total number of subsets of \(A \times B\) is \(2^{10} = 1024\).
- Subsets with at least 3 and at most 6 elements are given by: \[ \binom{10}{3} + \binom{10}{4} + \binom{10}{5} + \binom{10}{6}. \] - Calculate each term: \[ \binom{10}{3} = 120, \quad \binom{10}{4} = 210, \quad \binom{10}{5} = 252, \quad \binom{10}{6} = 210. \] - Summing these: \[ 120 + 210 + 252 + 210 = 792. \] Final Answer: The number of subsets is \(792\).