In Exercises 45-48, use combinations and permutations.
46. Five players on a basketball team must each choose one of the five players on the opposing team to defend. In how many ways can the players choose their defensive assignments?

Verified step by step guidance

1. Step 1: Recognize that this is a permutations problem because the order in which the defensive assignments are made matters. Each player on the basketball team is assigned to defend a specific player on the opposing team.
2. Step 2: Recall the formula for permutations, which is used when the order of selection matters. The formula is P(n, r) = n! / (n - r)!, where n is the total number of items to choose from, and r is the number of items being chosen.
3. Step 3: In this problem, there are 5 players on the basketball team and 5 players on the opposing team. Since all 5 players are being assigned, n = 5 and r = 5. Substitute these values into the permutation formula: P(5, 5) = 5! / (5 - 5)!.
4. Step 4: Simplify the expression. The factorial of 5 (5!) is calculated as 5 × 4 × 3 × 2 × 1. The factorial of 0 (0!) is defined as 1. Therefore, P(5, 5) = 5! / 1.
5. Step 5: Conclude that the total number of ways the players can choose their defensive assignments is equal to the value of 5!. This represents the number of permutations of 5 players.

Key Concepts

Permutations

Permutations refer to the different arrangements of a set of items where the order matters. In the context of the basketball question, each player choosing a defender from the opposing team represents a unique arrangement, as the specific choice of defender by each player affects the overall assignment.

Combinations

Combinations involve selecting items from a larger set where the order does not matter. While the basketball scenario primarily deals with permutations, understanding combinations is essential for grasping how selections can be made without regard to the order of choices, which can be relevant in different contexts.

Factorial

Factorial, denoted as n!, is the product of all positive integers up to n. It is a fundamental concept in counting arrangements and is used to calculate the total number of ways to arrange or select items. In the basketball problem, factorials help determine the total number of unique defensive assignments based on the players' choices.