In Exercises 15-18, determine whether the situation involves permutations, combinations, or neither. Explain your reasoning.
17. The number of ways 2 captains can be chosen from 28 players on a lacrosse team

Verified step by step guidance

1. Determine whether the order of selection matters in the problem. Since the problem is about choosing 2 captains and the order in which they are chosen does not matter (e.g., Captain A and Captain B is the same as Captain B and Captain A), this situation involves combinations.
2. Recall the formula for combinations: C(n, r) = n! / [(r!)(n - r)!], where n is the total number of items to choose from, and r is the number of items to choose.
3. Identify the values of n and r in this problem. Here, n = 28 (the total number of players) and r = 2 (the number of captains to be chosen).
4. Substitute the values of n and r into the combination formula: C(28, 2) = 28! / [(2!)(28 - 2)!].
5. Simplify the expression by canceling out the common factorial terms in the numerator and denominator, leaving you with C(28, 2) = (28 × 27) / 2!.

Key Concepts

Permutations

Permutations refer to the arrangement of items where the order matters. For example, if you are selecting a president and a vice president from a group, the order in which you select them is important, as different roles imply different arrangements.

Combinations

Combinations involve selecting items where the order does not matter. For instance, choosing 2 captains from a group of players means that selecting Player A and Player B is the same as selecting Player B and Player A; thus, the arrangement is irrelevant.

Choosing from a Group

When determining how to choose from a group, it's essential to identify whether the selection involves distinct roles or positions. In this case, since the captains are not assigned specific roles beyond being captains, the situation is a combination, as the order of selection does not affect the outcome.