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

Answer

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 1: Calculate the number of subsets containing two elements.
To find the number of ways to choose 2 elements from a set of 10 elements, we use the combination formula:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

Here, \( n = 10 \) and \( r = 2 \):

\[ \binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10!}{2! \cdot 8!} \]

### Step 2: Simplify the expression for two elements.
We can simplify this as follows:

\[ \binom{10}{2} = \frac{10 \times 9}{2 \times 1} = \frac{90}{2} = 45 \]

So, the number of subsets containing two elements is 45.

### Step 3: Calculate the number of subsets containing three elements.
Now, we calculate the number of ways to choose 3 elements from the same set:

\[ \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10!}{3! \cdot 7!} \]

### Step 4: Simplify the expression for three elements.
We can simplify this as follows:

\[ \binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = \frac{720}{6} = 120 \]

So, the number of subsets containing three elements is 120.

### Step 5: Add the two results together.
Now, we add the number of subsets containing two elements and the number of subsets containing three elements:

\[ \text{Total subsets} = \text{Subsets with 2 elements} + \text{Subsets with 3 elements} = 45 + 120 = 165 \]

### Conclusion
Thus, the total number of subsets of \( A \) containing either two or three elements is:

\[ \boxed{165} \]

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