If A and B are two sets such that `n(A)=2` and `n(B)=4`, then the total number of subsets of `AxxB` each having at least 3 elements are

Options

A. 219
B. 247
C. 239
D. 248

Answer

To solve the problem, we need to find the total number of subsets of the Cartesian product \( A \times B \) that have at least 3 elements. Here are the steps to arrive at the solution:

### Step 1: Determine the sizes of sets A and B
Given:
- \( n(A) = 2 \)
- \( n(B) = 4 \)

### Step 2: Calculate the size of the Cartesian product \( A \times B \)
The number of elements in the Cartesian product \( A \times B \) is given by:
\[ n(A \times B) = n(A) \times n(B) = 2 \times 4 = 8 \]

### Step 3: Calculate the total number of subsets of \( A \times B \)
The total number of subsets of a set with \( n \) elements is given by \( 2^n \). Therefore, the total number of subsets of \( A \times B \) is:
\[ 2^{n(A \times B)} = 2^8 = 256 \]

### Step 4: Calculate the number of subsets with less than 3 elements
We need to find the number of subsets that have less than 3 elements, which includes subsets with 0, 1, or 2 elements.

- **Subsets with 0 elements**: There is 1 subset (the empty set).
\[ \binom{8}{0} = 1 \]

- **Subsets with 1 element**: The number of ways to choose 1 element from 8.
\[ \binom{8}{1} = 8 \]

- **Subsets with 2 elements**: The number of ways to choose 2 elements from 8.
\[ \binom{8}{2} = \frac{8 \times 7}{2 \times 1} = 28 \]

### Step 5: Sum the subsets with less than 3 elements
Now, we add the number of subsets with 0, 1, and 2 elements:
\[ \text{Total subsets with less than 3 elements} = \binom{8}{0} + \binom{8}{1} + \binom{8}{2} = 1 + 8 + 28 = 37 \]

### Step 6: Calculate the number of subsets with at least 3 elements
To find the number of subsets with at least 3 elements, we subtract the number of subsets with less than 3 elements from the total number of subsets:
\[ \text{Subsets with at least 3 elements} = 256 - 37 = 219 \]

### Final Answer
The total number of subsets of \( A \times B \) that have at least 3 elements is:
\[ \boxed{219} \]