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Chapter 11: Problem 72

I used the formula for \({ }_{n} C_{r}\) to determine the number of possible outcomes when ranking five politicians from most admired to least admired.

Short Answer

Expert verified
Therefore, there are 120 ways the politicians can be ranked from most admired to least admired.

Step by step solution

01

Understand the Problem

The problem involves finding the permutations of a set of 5 politicians, ranked from most admired to least admired.
02

Use of Permutation Formula

Instead of using the combination formula (\({ }_{n} C_{r}\)), the permutation formula (\({ }_{n} P_{r}\)) should be used, where \(n\) is the total number of politicians and \(r\) is the number of rankings. Here, \(n=5\) and \(r=5\).
03

Apply the Permutation Formula

The formula for permutations is \({ }_{n} P_{r} = \frac{n!}{(n-r)!}\). Since we are ranking all politicians, \(r\) is equal to \(n\), which means (n-r)! becomes 0!. Note that 0! is defined as 1.
04

Calculate the Permutation

Applying the formula gives \({ }_{5} P_{5} = \frac{5!}{(5-5)!} = \frac{5!}{0!} = 5! = 5 * 4 * 3 * 2 * 1 = 120\)

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