Let A={1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Then the number of subsets of A containing exactly two elements is

Options

A. 20
B. 40
C. 45
D. 90

Explanation

To find the number of subsets of the set \( A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \) that contain exactly two elements, we can use the concept of combinations.

### Step-by-Step Solution:

1. **Identify the Total Number of Elements**:
The set \( A \) contains 10 elements.

2. **Choose 2 Elements from 10**:
We need to choose 2 elements from these 10 elements. The number of ways to choose \( r \) elements from a set of \( n \) elements is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, \( n = 10 \) and \( r = 2 \).

3. **Substitute the Values into the Formula**:
\[ \binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10!}{2! \cdot 8!} \]

4. **Simplify the Factorials**:
Since \( 10! = 10 \times 9 \times 8! \), we can cancel \( 8! \) in the numerator and denominator:
\[ \binom{10}{2} = \frac{10 \times 9 \times 8!}{2! \times 8!} = \frac{10 \times 9}{2!} \]

5. **Calculate \( 2! \)**:
\[ 2! = 2 \times 1 = 2 \]

6. **Final Calculation**:
\[ \binom{10}{2} = \frac{10 \times 9}{2} = \frac{90}{2} = 45 \]

Thus, the number of subsets of \( A \) containing exactly two elements is \( 45 \).