Chapter 10: Problem 38
Tell whether the question can be answered using permutations or combinations. Explain your reasoning. Then answer the question. Ten students are auditioning for 3 different roles in a play. In how many ways can the 3 roles be filled?
Short Answer
Expert verified
There are 720 ways the 3 roles can be filled by the 10 students.
Step by step solution
01
Identify the Problem Type
Understand that the problem is about permutations since the order of the roles matters and they are distinct.
02
Apply the Permutation Formula
In permutations, the formula to calculate the number of ways \(r\) items can be selected out of \(n\) items is \( nPr = \frac{n!}{(n-r)!} \). Here n represents the total number of students (10) and r represents the different roles (3). Factorial '!' is the product of all positive integers up to that number.
03
Calculate the Result
Substitute n and r into the permutation formula: \( 10P3 = \frac{10!}{(10-3)!} \) , which simplifies to \( = \frac{10*9*8*7!}{7!}\). The factorials cancel out, leaving \(10*9*8\), which equals to 720.
04
Interpret the Result
This means that there are 720 different ways that the 3 different roles can be filled by the 10 students.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Permutation Formula
Permutations are used when the order of selection is important. In this context, we have three roles to fill, and the order or assignment of these roles matters. This scenario requires the application of the permutation formula:
\[ nPr = \frac{n!}{(n-r)!} \]
This formula calculates permutations by determining how to order sequences of items. The notation \(nPr\) represents choosing \(r\) items from \(n\) items, where each selection sequence has a unique order that matters. If you want to find out how the permutation formula applies to our problem, we substitute \(n\) with 10 (students) and \(r\) with 3 (roles). This means we're calculating how to arrange these students into these roles in sequence which is unique and varied.Using this approach, every arrangement of individuals into roles counts as a distinct outcome.
\[ nPr = \frac{n!}{(n-r)!} \]
This formula calculates permutations by determining how to order sequences of items. The notation \(nPr\) represents choosing \(r\) items from \(n\) items, where each selection sequence has a unique order that matters. If you want to find out how the permutation formula applies to our problem, we substitute \(n\) with 10 (students) and \(r\) with 3 (roles). This means we're calculating how to arrange these students into these roles in sequence which is unique and varied.Using this approach, every arrangement of individuals into roles counts as a distinct outcome.
Factorial
The factorial often represented as \(!\), is a mathematical operation crucial for permutations and combinations. It involves multiplying a sequence of descending positive integers. For example, \(5!\) is \(5 \times 4 \times 3 \times 2 \times 1\). Factorials grow rapidly and become large numbers quickly.
In our problem, we utilize factorials in the permutation formula. When we substitute 10 for the total number of students, we compute \(10!\). This operation becomes crucial when using formulas that require sequential subtraction, as seen in the formula:\[ \frac{n!}{(n-r)!} \]
Here, \((n-r)!\) effectively removes the sequence of orders involving only the selected roles. It systematically zeroes out anything not related to our actual selection of roles.
In our problem, we utilize factorials in the permutation formula. When we substitute 10 for the total number of students, we compute \(10!\). This operation becomes crucial when using formulas that require sequential subtraction, as seen in the formula:\[ \frac{n!}{(n-r)!} \]
Here, \((n-r)!\) effectively removes the sequence of orders involving only the selected roles. It systematically zeroes out anything not related to our actual selection of roles.
Role Assignment
Role assignment is a key part of why we use permutations instead of combinations. Assigning roles means each selected student takes up a unique, distinguishable place. In our exercise, three different roles indicate that each role has different duties or positions.
Consider auditioning for a play: you wouldn't randomly place characters without regard to the narrative (or order). Since roles have different requirements or scenes, who assumes each role defines their placement in the play and is not interchangeable. This makes it a permutation problem where each assignment needs full consideration of its position. Every different set of role assignments creates a new unique arrangement or sequence, precisely what permutations account for.
Consider auditioning for a play: you wouldn't randomly place characters without regard to the narrative (or order). Since roles have different requirements or scenes, who assumes each role defines their placement in the play and is not interchangeable. This makes it a permutation problem where each assignment needs full consideration of its position. Every different set of role assignments creates a new unique arrangement or sequence, precisely what permutations account for.
Combinations
Though not directly used in our exercise, understanding combinations can clarify why permutations are the proper choice. Combinations involve selecting items where order does not matter. This is illustrated through the combination formula:
\[ nCr = \frac{n!}{r!(n-r)!} \]
In combinations, who is selected does not change the group, only that they have been selected is what matters. For example, choosing team members in sports generally involves combinations, as the order they are picked doesn't impact their spot on the team.Our role assignment demands specific place-holding with a definitive order. This ignores combinations specifically because each student filling a role is crucial to that specific role, not just that they’re chosen. Mixing this concept often leads to incorrect answers where order variations are overlooked, reducing total possible outcomes.
\[ nCr = \frac{n!}{r!(n-r)!} \]
In combinations, who is selected does not change the group, only that they have been selected is what matters. For example, choosing team members in sports generally involves combinations, as the order they are picked doesn't impact their spot on the team.Our role assignment demands specific place-holding with a definitive order. This ignores combinations specifically because each student filling a role is crucial to that specific role, not just that they’re chosen. Mixing this concept often leads to incorrect answers where order variations are overlooked, reducing total possible outcomes.