If G = {1, 2, 3, 4, 5}, how many subsets have exactly 3 elements?

If G = {1, 2, 3, 4, 5}, how many subsets have exactly 3 elements?

Correct Answer: 10

Step 1: Understand that a subset is a selection of elements from a set.
Step 2: Identify the set G, which contains 5 elements: {1, 2, 3, 4, 5}.
Step 3: We want to find subsets that have exactly 3 elements.
Step 4: Use the combination formula C(n, k) to find the number of ways to choose k elements from n elements.
Step 5: In this case, n = 5 (the total number of elements in G) and k = 3 (the number of elements we want in each subset).
Step 6: The combination formula is C(n, k) = n! / (k! * (n - k)!).
Step 7: Calculate C(5, 3) using the formula: C(5, 3) = 5! / (3! * (5 - 3)!) = 5! / (3! * 2!).
Step 8: Calculate the factorials: 5! = 120, 3! = 6, and 2! = 2.
Step 9: Substitute the factorials into the formula: C(5, 3) = 120 / (6 * 2) = 120 / 12 = 10.
Step 10: Conclude that there are 10 subsets of G that have 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.