Combinatorics as Problem-Solving Exercises

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 threestop 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)!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.
Note: A permutation function is included on most modern scientific calculators. You
may find it in the menu of functions under “probability.”
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 15minute 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.