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

Options

A. 220
B. 219
C. 211
D. 256

Explanation

To solve the problem step by step, let's break down the solution:

### Step 1: Determine the number of elements in the Cartesian product A x B
Given:
- Set A has 2 elements.
- Set B has 4 elements.

The number of elements in the Cartesian product \( A \times B \) is calculated as:
\[ |A \times B| = |A| \times |B| = 2 \times 4 = 8 \]

### Step 2: 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 3: Calculate the number of subsets with fewer than 3 elements
We need to find the number of subsets that have 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 4: Sum the subsets with fewer than 3 elements
Now, we add the number of 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 5: 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:
\[ \text{Subsets with 3 or more elements} = 256 - 37 = 219 \]

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

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