If D = {1, 2, 3, 4, 5, 6}, how many subsets contain exactly 3 elements?

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Question: If D = {1, 2, 3, 4, 5, 6}, how many subsets contain exactly 3 elements?

Options:

Correct Answer: 20

Solution:

The number of ways to choose 3 elements from a set of 6 is given by the combination formula C(n, r) = n! / (r!(n-r)!). Here, C(6, 3) = 20.

If D = {1, 2, 3, 4, 5, 6}, how many subsets contain exactly 3 elements?

If D = {1, 2, 3, 4, 5, 6}, how many subsets contain exactly 3 elements?

Step 1: Identify the set D, which contains the elements {1, 2, 3, 4, 5, 6}.
Step 2: Determine how many elements you want in each subset. In this case, we want subsets with exactly 3 elements.
Step 3: Use the combination formula C(n, r) to find the number of ways to choose r elements from a set of n elements. The formula is C(n, r) = n! / (r!(n-r)!).
Step 4: Plug in the values into the formula. Here, n = 6 (the total number of elements in set D) and r = 3 (the number of elements we want in each subset).
Step 5: Calculate C(6, 3) using the formula: C(6, 3) = 6! / (3!(6-3)!) = 6! / (3! * 3!).
Step 6: Calculate the factorials: 6! = 720, 3! = 6, so C(6, 3) = 720 / (6 * 6) = 720 / 36 = 20.
Step 7: Conclude that there are 20 different subsets that contain exactly 3 elements.

Combinations – The concept of combinations is used to determine the number of ways to choose a subset of items from a larger set without regard to the order of selection.