Given that b = {1, 2, 3, 4}, how many subsets have exactly two elements?

To find how many subsets contain exactly two elements from the set b = {1, 2, 3, 4}, we can use the concept of combinations from combinatorial mathematics. The formula for combinations is given by:

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

Where:

• n = total number of elements in the set,
• r = number of elements to choose,
• ! = factorial, which is the product of all positive integers up to that number.

In this case, we have:

• n = 4 (since there are four elements: 1, 2, 3, 4),
• r = 2 (since we want subsets of exactly two elements).

Substituting these values into the formula gives us:

C(4, 2) = 4! / (2!(4 - 2)!) = 4! / (2! * 2!)

Calculating the factorials:

• 4! = 24,
• 2! = 2, hence 2! * 2! = 2 * 2 = 4.

Now substituting these values back:

C(4, 2) = 24 / 4 = 6

Thus, there are 6 subsets of the set b = {1, 2, 3, 4} that have exactly two elements. The subsets are:

• {1, 2}
• {1, 3}
• {1, 4}
• {2, 3}
• {2, 4}
• {3, 4}