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Chapter 11: Problem 60

From a pool of 7 secretaries, 3 are selected to be assigned to 3 managers, 1 secretary to each manager. In how many ways can this be done?

Short Answer

Expert verified
210 ways.

Step by step solution

01

Understand the problem

From 7 secretaries, 3 need to be selected and assigned to 3 managers. Each manager gets one secretary.
02

Calculate the number of ways to choose 3 secretaries from 7

Use combination formula to choose 3 secretaries from 7, without considering the arrangement yet. The formula for combination is \({{7}\brace{3}} = \frac{7!}{3!(7-3)!} = 35\).
03

Calculate the number of ways to assign the 3 secretaries to 3 managers

After selecting 3 secretaries, they need to be assigned to 3 managers. This can be done in \(3!\) ways. \(3! = 6\).
04

Calculate the total number of ways

Multiply the number of combinations by the number of permutations: \({{7}\brace{3}} \times 3! = 35 \times 6 = 210\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

combination formula
The combination formula helps us determine the number of ways to choose a subset of items from a larger set, where the order in which the items are chosen does not matter.

The combination formula is denoted as \({n \brace k}\), and is defined as: \[ \[\begin{equation} {n \brace k} = \frac{n!}{k! (n-k)!} \end{equation}\] \] Here, \(n\) is the total number of items, and \(k\) is the number of items to choose.

For instance, in the given problem, we need to choose 3 secretaries out of 7, so we use the formula \( {7 \brace 3} = \frac{7!}{3!(7-3)!} = 35 \). This means there are 35 ways to choose 3 secretaries from a total of 7.
permutation
Permutation considers the arrangement of items where the order does matter.

For permutations, the formula is given as: \[ \[\begin{equation} P(n, k) = \frac{n!}{(n-k)!} \end{equation}\] \] Here, \(n\) is the total number of items, and \(k\) is the number of items to arrange.

In the exercise provided, once we have selected 3 secretaries, we need to assign them to 3 managers. This can be done in \(3!\) ways, which equals 6. Each unique way of assigning the secretaries is considered a different permutation.
factorial
The factorial is a core concept in both combinations and permutations. It is denoted as \(n!\), and represents the product of all positive integers up to \(n\). For example \(5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120\).

Factorials are instrumental in calculating combinations and permutations because they help simplify and manage large numbers. In the above problem, we used factorials to determine both the number of ways to choose and assign secretaries. For the combination, \(7! = 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 5040\). To get the number of permutations, we calculate \(3! = 3 \cdot 2 \cdot 1 = 6\).

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