Textbook Question
1. When you calculate the number of permutations of n distinct objects taken r at a time, what are you counting? Give an example.
4. Probability
Counting
1:11 minutesProblem 3.4.17
Textbook Question
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 guidance1
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.
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.
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).
Substitute the values of n and r into the combination formula: C(28, 2) = 28! / [(2!)(28 - 2)!].
Simplify the expression by canceling out the common factorial terms in the numerator and denominator, leaving you with C(28, 2) = (28 × 27) / 2!.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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