If A and B two sets containing 2 elements and 4 elements, respectively. Then, the number of subsets of `A xx B` having 3 or more elements, is

Explanation

To solve the problem, we need to find the number of subsets of the Cartesian product \( A \times B \) that have 3 or more elements. Let's break this down step by step.

### Step 1: Determine the sizes of sets A and B
Given:
- Set \( A \) has 2 elements.
- Set \( B \) has 4 elements.

### Step 2: Calculate the size of the Cartesian product \( A \times B \)
The size of the Cartesian product \( A \times B \) is given by the product of the sizes of the two sets:
\[ |A \times B| = |A| \times |B| = 2 \times 4 = 8 \]

### 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, for \( A \times B \):
\[ \text{Total subsets} = 2^{|A \times B|} = 2^8 = 256 \]

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

1. **Number of subsets with 0 elements (empty set)**:
\[ 1 \text{ (the empty set)} \]

2. **Number of subsets with 1 element**:
\[ \binom{8}{1} = 8 \]

3. **Number of subsets with 2 elements**:
\[ \binom{8}{2} = \frac{8 \times 7}{2 \times 1} = 28 \]

### Step 5: Calculate the total number of subsets with fewer than 3 elements
Now, we sum the subsets with 0, 1, and 2 elements:
\[ \text{Total subsets with fewer than 3 elements} = 1 + 8 + 28 = 37 \]

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

### Final Answer
The number of subsets of \( A \times B \) having 3 or more elements is \( \boxed{219} \).

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