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

Options

A. 45
B. 120
C. 165
D. 330

Explanation

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

### Step-by-Step Solution:

1. **Identify the Total Elements in Set A**:
The set \( A \) has 10 elements: \( \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \).

2. **Calculate the Number of 2-Element Subsets**:
The number of ways to choose 2 elements from 10 can be calculated using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, \( n = 10 \) and \( r = 2 \):
\[ \binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45 \]

3. **Calculate the Number of 3-Element Subsets**:
Similarly, the number of ways to choose 3 elements from 10 is:
\[ \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]

4. **Add the Two Results**:
To find the total number of subsets containing either two or three elements, we add the two results:
\[ \text{Total subsets} = \binom{10}{2} + \binom{10}{3} = 45 + 120 = 165 \]

5. **Conclusion**:
Therefore, the number of subsets of \( A \) containing either two or three elements is \( 165 \).

### Final Answer:
The number of subsets of \( A \) containing two or three elements is \( 165 \).