Question Analysis:
We are given a set A with 10 elements. We need to determine the number of subsets of A that contain exactly two elements.

Solution:
To solve this problem, we can use the concept of combinations.

Step 1: Find the number of ways to choose 2 elements from a set of 10.
To choose 2 elements from a set of 10, we can use the combination formula:

C(n, r) = n! / (r! * (n-r)!)

where n is the total number of elements in the set and r is the number of elements to be chosen.

In this case, n = 10 and r = 2. Plugging these values into the formula, we get:

C(10, 2) = 10! / (2! * (10-2)!)
= 10! / (2! * 8!)
= (10 * 9) / (2 * 1)
= 45

So, there are 45 ways to choose 2 elements from a set of 10.

Step 2: Find the number of subsets containing exactly 2 elements.
To find the number of subsets containing exactly 2 elements, we can use the formula:

Number of subsets = 2^n

where n is the number of elements in the set. In this case, n = 10. Plugging this value into the formula, we get:

Number of subsets = 2^10 = 1024

However, this includes subsets of all sizes, not just those containing exactly 2 elements.

Step 3: Subtract the number of subsets containing more or less than 2 elements.
To find the number of subsets containing exactly 2 elements, we need to subtract the number of subsets containing more or less than 2 elements from the total number of subsets.

Number of subsets containing more than 2 elements:
There are C(10, 3) ways to choose 3 elements from a set of 10. Using the combination formula, we get:

C(10, 3) = 10! / (3! * (10-3)!)
= (10 * 9 * 8) / (3 * 2 * 1)
= 120

Number of subsets containing less than 2 elements:
There is only 1 way to choose 1 element from a set of 10.

So, the total number of subsets containing more or less than 2 elements is:

120 + 1 = 121

Finally, we subtract the number of subsets containing more or less than 2 elements from the total number of subsets:

Number of subsets containing exactly 2 elements = Total number of subsets - Number of subsets containing more or less than 2 elements
= 1024 - 121
= 903

Therefore, the number of subsets of A containing exactly two elements is 903.