Refer back to the Learning Activities titled “A Formula for Permutations” and “A Formula for Combinations.” Explain how the Fundamental Counting Principle is used each time the Permutation Formula (nPr) or the Combination Formula (nCr) is applied. Would these be considered independent or dependent events? Or, is it inappropriate to be concerned about whether it is independent/dependent? Explain your thinking on the question.

A Formula for Permutations

Introduction

Formulas can be a very efficient way of solving problems. For the permutation formula introduced in this section, identify all the relevant variables. How does the formula function? How does it align to other methods of solving permutation problems?

Max needs to figure out how many possible route options he has going from one city to another on his two-week vacation in Europe. He has 10 cities on his itinerary and would like to visit at least three of them. How many possible route options does Max have, if he goes to three of the 10 cities during his vacation?

Now suppose there are 10 locations and only three stops can be made over a given time period. In this case, how many ordered three-stop routes out of 10 possible stops can be considered? We could use the method:

, or we can use the permutation rule where k items are to be selected from n available items. When order matters, this idea is notated where

nPk=n!(n−k)!

To solve this particular routing problem, use n = 10 and k = 3.

That is, out of 10 locations, there are 720 possible three-location routes.

Moreover, you can see that this does conform to the fundamental counting principle. There are 10 locations to choose from for the first stop, then nine for the second, and eight for the third.

Try This!

A company employs 24 engineers, and an event requires four 15-minute presentations given by engineers. How many different lineups for presentations are possible?

(c) Thinkstock

Solution: Here we need to select four engineers out of 24, where order is relevant. Therefore, we will use the permutation rule,

where n = 24 and r = 4.

Answer: 255,024 lineups

Permutations

The following video tutorial explains how to evaluate factorials, use permutations to solve problems, and determine the number of permutations with indistinguishable items.

(Mathispower4u, 2010)

Answer preview

The fundamental counting principle (FCP) explains that where independent events occur in a series a₁, a₂, a₃, a₄  … a the number of all the events that occur is defined by a₁* a₂* a₃* a₄ ..a_n. For instance, how many times can someone answer yes or no for four questions? Two events can be selected four times, and hence FCP defines the value; . This indicates that the selection of the events can be repeated, and the order does not matter.

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