Let A and B be too sets containing four and two elements respectively then the number of subsets of set `AxxB` having atleast 3 elements is

Options

A. 275
B. 510
C. 219
D. 256

Explanation

To solve the problem, we need to find the number of subsets of the Cartesian product of two sets \( A \) and \( B \) that have at least 3 elements. Let's break it down step by step.

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

### Step 2: Calculate the cardinality of the Cartesian product \( A \times B \)
The number of elements in the Cartesian product \( A \times B \) is given by:
\[ |A \times B| = |A| \times |B| = 4 \times 2 = 8 \]
So, the set \( A \times B \) has 8 elements.

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

### Step 4: Calculate the number of subsets with fewer than 3 elements
We need to find the number of subsets that have at least 3 elements. To do this, we first calculate the number of subsets with 0, 1, and 2 elements.

1. **Subsets with 0 elements**:
\[ \binom{8}{0} = 1 \]

2. **Subsets with 1 element**:
\[ \binom{8}{1} = 8 \]

3. **Subsets with 2 elements**:
\[ \binom{8}{2} = \frac{8!}{2!(8-2)!} = \frac{8 \times 7}{2 \times 1} = 28 \]

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

### 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 fewer than 3 elements from the total number of subsets:
\[ \text{Subsets with at least 3 elements} = \text{Total subsets} - \text{Subsets with fewer than 3 elements} \]
\[ = 256 - 37 = 219 \]

### Final Answer
The number of subsets of the set \( A \times B \) having at least 3 elements is **219**.

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