If `A={1,2,3}` and `B={1,2}` and `C={4,5,6}`, then what is the number of elements in the set `AxxBxxC`?

Options

A. 8
B. 9
C. 15
D. 18

Explanation

To find the number of elements in the set \( A \times B \times C \), we can use the formula for the Cartesian product of sets. The number of elements in the Cartesian product of multiple sets is the product of the number of elements in each set.

### Step-by-step Solution:

1. **Identify the sets and their elements**:
- Set \( A = \{1, 2, 3\} \) has 3 elements.
- Set \( B = \{1, 2\} \) has 2 elements.
- Set \( C = \{4, 5, 6\} \) has 3 elements.

2. **Count the number of elements in each set**:
- Number of elements in set \( A \) is \( |A| = 3 \).
- Number of elements in set \( B \) is \( |B| = 2 \).
- Number of elements in set \( C \) is \( |C| = 3 \).

3. **Use the formula for the Cartesian product**:
The number of elements in the Cartesian product \( A \times B \times C \) is given by:
\[ |A \times B \times C| = |A| \times |B| \times |C| \]

4. **Substitute the values**:
\[ |A \times B \times C| = 3 \times 2 \times 3 \]

5. **Calculate the product**:
- First, calculate \( 3 \times 2 = 6 \).
- Then, calculate \( 6 \times 3 = 18 \).

6. **Conclusion**:
Therefore, the number of elements in the set \( A \times B \times C \) is \( 18 \).

### Final Answer:
The number of elements in the set \( A \times B \times C \) is \( 18 \).

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