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Contemporary Mathematics

7.2 Permutations

Contemporary Mathematics7.2 Permutations

Table of contents
  1. Preface
  2. 1 Sets
    1. Introduction
    2. 1.1 Basic Set Concepts
    3. 1.2 Subsets
    4. 1.3 Understanding Venn Diagrams
    5. 1.4 Set Operations with Two Sets
    6. 1.5 Set Operations with Three Sets
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  3. 2 Logic
    1. Introduction
    2. 2.1 Statements and Quantifiers
    3. 2.2 Compound Statements
    4. 2.3 Constructing Truth Tables
    5. 2.4 Truth Tables for the Conditional and Biconditional
    6. 2.5 Equivalent Statements
    7. 2.6 De Morgan’s Laws
    8. 2.7 Logical Arguments
    9. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  4. 3 Real Number Systems and Number Theory
    1. Introduction
    2. 3.1 Prime and Composite Numbers
    3. 3.2 The Integers
    4. 3.3 Order of Operations
    5. 3.4 Rational Numbers
    6. 3.5 Irrational Numbers
    7. 3.6 Real Numbers
    8. 3.7 Clock Arithmetic
    9. 3.8 Exponents
    10. 3.9 Scientific Notation
    11. 3.10 Arithmetic Sequences
    12. 3.11 Geometric Sequences
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  5. 4 Number Representation and Calculation
    1. Introduction
    2. 4.1 Hindu-Arabic Positional System
    3. 4.2 Early Numeration Systems
    4. 4.3 Converting with Base Systems
    5. 4.4 Addition and Subtraction in Base Systems
    6. 4.5 Multiplication and Division in Base Systems
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  6. 5 Algebra
    1. Introduction
    2. 5.1 Algebraic Expressions
    3. 5.2 Linear Equations in One Variable with Applications
    4. 5.3 Linear Inequalities in One Variable with Applications
    5. 5.4 Ratios and Proportions
    6. 5.5 Graphing Linear Equations and Inequalities
    7. 5.6 Quadratic Equations with Two Variables with Applications
    8. 5.7 Functions
    9. 5.8 Graphing Functions
    10. 5.9 Systems of Linear Equations in Two Variables
    11. 5.10 Systems of Linear Inequalities in Two Variables
    12. 5.11 Linear Programming
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  7. 6 Money Management
    1. Introduction
    2. 6.1 Understanding Percent
    3. 6.2 Discounts, Markups, and Sales Tax
    4. 6.3 Simple Interest
    5. 6.4 Compound Interest
    6. 6.5 Making a Personal Budget
    7. 6.6 Methods of Savings
    8. 6.7 Investments
    9. 6.8 The Basics of Loans
    10. 6.9 Understanding Student Loans
    11. 6.10 Credit Cards
    12. 6.11 Buying or Leasing a Car
    13. 6.12 Renting and Homeownership
    14. 6.13 Income Tax
    15. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  8. 7 Probability
    1. Introduction
    2. 7.1 The Multiplication Rule for Counting
    3. 7.2 Permutations
    4. 7.3 Combinations
    5. 7.4 Tree Diagrams, Tables, and Outcomes
    6. 7.5 Basic Concepts of Probability
    7. 7.6 Probability with Permutations and Combinations
    8. 7.7 What Are the Odds?
    9. 7.8 The Addition Rule for Probability
    10. 7.9 Conditional Probability and the Multiplication Rule
    11. 7.10 The Binomial Distribution
    12. 7.11 Expected Value
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  9. 8 Statistics
    1. Introduction
    2. 8.1 Gathering and Organizing Data
    3. 8.2 Visualizing Data
    4. 8.3 Mean, Median and Mode
    5. 8.4 Range and Standard Deviation
    6. 8.5 Percentiles
    7. 8.6 The Normal Distribution
    8. 8.7 Applications of the Normal Distribution
    9. 8.8 Scatter Plots, Correlation, and Regression Lines
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  10. 9 Metric Measurement
    1. Introduction
    2. 9.1 The Metric System
    3. 9.2 Measuring Area
    4. 9.3 Measuring Volume
    5. 9.4 Measuring Weight
    6. 9.5 Measuring Temperature
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  11. 10 Geometry
    1. Introduction
    2. 10.1 Points, Lines, and Planes
    3. 10.2 Angles
    4. 10.3 Triangles
    5. 10.4 Polygons, Perimeter, and Circumference
    6. 10.5 Tessellations
    7. 10.6 Area
    8. 10.7 Volume and Surface Area
    9. 10.8 Right Triangle Trigonometry
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  12. 11 Voting and Apportionment
    1. Introduction
    2. 11.1 Voting Methods
    3. 11.2 Fairness in Voting Methods
    4. 11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem
    5. 11.4 Apportionment Methods
    6. 11.5 Fairness in Apportionment Methods
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  13. 12 Graph Theory
    1. Introduction
    2. 12.1 Graph Basics
    3. 12.2 Graph Structures
    4. 12.3 Comparing Graphs
    5. 12.4 Navigating Graphs
    6. 12.5 Euler Circuits
    7. 12.6 Euler Trails
    8. 12.7 Hamilton Cycles
    9. 12.8 Hamilton Paths
    10. 12.9 Traveling Salesperson Problem
    11. 12.10 Trees
    12. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  14. 13 Math and...
    1. Introduction
    2. 13.1 Math and Art
    3. 13.2 Math and the Environment
    4. 13.3 Math and Medicine
    5. 13.4 Math and Music
    6. 13.5 Math and Sports
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  15. A | Co-Req Appendix: Integer Powers of 10
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index
Three swimmers are racing in separate lanes in a swimming pool. The swimmer in the top lane is in first place, the swimmer in the bottom lane is in second place, and the swimmer in the middle lane is in third place.
Figure 7.7 We can use permutations to calculate the number of different orders of finish in an Olympic swimming heat. (credit: “London 2012 Olympics Park Stratford London” by Gary Bembridge/Flickr, CC BY 2.0)

Learning Objectives

After completing this section, you should be able to:

  1. Use the Multiplication Rule for Counting to determine the number of permutations.
  2. Compute expressions containing factorials.
  3. Compute permutations.
  4. Apply permutations to solve problems.

Swimming events are some of the most popular events at the summer Olympic Games. In the finals of each event, 8 swimmers compete at the same time, making for some exciting finishes. How many different orders of finish are possible in these events? In this section, we’ll extend the Multiplication Rule for Counting to help answer questions like this one, which relate to permutations. A permutation is an ordered list of objects taken from a given population. The length of the list is given, and the list cannot contain any repeated items.

Applying the Multiplication Rule for Counting to Permutations

In the case of the swimming finals, one possible permutation of length 3 would be the list of medal winners (first, second, and third place finishers). A permutation of length 8 would be the full order of finish (first place through eighth place). Let’s use the Multiplication Rule for Counting to figure out how many of each of these permutations there are.

Example 7.4

Using the Multiplication Rule for Counting to Find the Number of Permutations

The final heat of Olympic swimming events features 8 swimmers (or teams of swimmers).

  1. How many different podium placements (first place, second place, and third place) are possible?
  2. How many different complete orders of finish (first place through eighth place) are possible?

Your Turn 7.4

1.
You have a hand of 5 cards (that happen to create what’s called a royal flush in the game of poker): 10 , J , Q , K , and A . Into how many different orders can you put those cards?

Factorials

The pattern we see in Example 7.4 occurs commonly enough that we have a name for it: factorial.
For any positive whole number nn, we define the factorial of nn (denoted n!n! and read "nn factorial") to be the product of every whole number less than or equal to nn. We also define 0! to be equal to one. We will use factorials in a couple of different contexts, so let's get some practice doing computations with them.

Example 7.5

Computing Factorials

Compute the following:

  1. 4!4!
  2. 8!6!8!6!
  3. 9!3!4!9!3!4!

Your Turn 7.5

Compute the following:
1.
6 !
2.
12 ! 10 !
3.
8 ! 4 ! 4 !

Permutations

As we’ve seen, factorials can pop up when we’re computing permutations. In fact, there is a formula that we can use to make that connection explicit. Let’s define some notation first. If we have a collection of nn objects and we wish to create an ordered list of rr of the objects (where 1rn1rn), we’ll call the number of those permutations nPrnPr (read “the number of permutations of nn objects taken rr at a time”). We formalize the formula we'll use to compute permutations below.

FORMULA

nPr=n!(nr)!nPr=n!(nr)!

If you wondered why we defined 0!=10!=1 earlier, it was to make formulas like this one work; if we have nn objects and want to order all of them (so, we want the number of permutations of nn objects taken nn at a time), we get nPn=n!(nn)!=n!0!=n!1=n!nPn=n!(nn)!=n!0!=n!1=n!. Next, we’ll get some practice computing these permutations.

Example 7.6

Computing Permutations

Find the following numbers:

  1. The number of permutations of 12 objects taken 3 at a time
  2. The number of permutations of 8 objects taken 5 at a time
  3. The number of permutations of 32 objects taken 2 at a time

Your Turn 7.6

Find the following numbers:
1.
The number of permutations of 6 objects taken 2 at a time
2.
The number of permutations of 14 objects taken 4 at a time
3.
The number of permutations of 19 objects taken 3 at a time

Example 7.7

Applying Permutations

  1. A high school graduating class has 312 students. The top student is declared valedictorian, and the second-best is named salutatorian. How many possible outcomes are there for the valedictorian and salutatorian?
  2. In the card game blackjack, the dealer’s hand of 2 cards is dealt with 1 card faceup and 1 card facedown. If the game is being played with a single deck of (52) cards, how many possible hands could the dealer get?
  3. The University Combinatorics Club has 3 officers: president, vice president, and treasurer. If there are 18 members of the club, how many ways are there to fill the officer positions?

Your Turn 7.7

1.
One of the big draws at this year’s state fair is the pig race. There are 15 entrants, and prizes are given to the top three finishers. How many different combinations of top-three finishes could there be?

Who Knew?

Very Big Permutations

Permutations involving relatively small sets of objects can get very big, very quickly. A standard deck contains 52 cards. So, the number of different ways to shuffle the cards—in other words, the number of permutations of 52 objects taken 52 at a time—is 52!8×106752!8×1067 (written out, that’s an 8 followed by 67 zeroes). The estimated age of the universe is only about 4×10174×1017 seconds. So, if a very bored all-powerful being started shuffling cards at the instant the universe began, it would have to have averaged at least 8×10674×10172×10508×10674×10172×1050 shuffles per second since the beginning of time to have covered every possible arrangement of a deck of cards. That means the next time you pick up a deck of cards and give it a good shuffle, it’s almost certain that the particular arrangement you created has never been created before and likely never will be created again.

Check Your Understanding

6.
Compute 5!.
7.
Compute 10 ! 7 ! 3 ! .
8.
Compute 12 P 3 .
9.
Compute 8 P 4 .
10.
The standard American edition of the board game Monopoly has a deck of 15 orange Chance cards. In how many different ways could the first 4 Chance cards drawn in a game appear?
For the following exercises, give a whole number that’s equal to the given expression.
1 .
3!
2 .
9!
3 .
7 ! 2 ! 2 ! 3 !
4 .
8 ! 5 ! 2 !
5 .
21 ! 18 ! 2 !
6 .
28 ! 26 ! 2 !
7 .
34 ! 30 ! 3 !
8 .
17 ! 12 ! 5 !
9 .
4 P 3
10 .
7 P 5
11 .
12 P 10
12 .
14 P 10
13 .
10 P 8
14 .
15 P 11
The following exercises are about the card game euchre, which uses a partial standard deck of cards: It only has the cards with ranks 9, 10, J, Q, K, and A for a total of 24 cards. Some variations of the game use the 8s or the 7s and 8s, but we’ll stick with the 24-card version.
15 .
A euchre hand contains 5 cards. How many ways are there to receive a 5-card hand (where the order in which the cards are received matters, i.e., 9 , J , K , 9 , 10 is different from 9 J , 9 , K , 10 ?
16 .
After all 4 players get their hands, the remaining 4 cards are placed facedown in the center of the table. How many arrangements of 4 cards are there from this deck?
17 .
Euchre is played with partners. How many ways are there for 2 partners to receive 5-card hands (where the order in which the cards are received matters)?
18 .
How many different arrangements of the full euchre deck are possible (i.e., how many different shuffles are there)?
The following exercises involve a horse race with 13 entrants.
19 .
How many possible complete orders of finish are there?
20 .
An exacta bet is one where the player tries to predict the top two finishers in order. How many possible exacta bets are there for this race?
21 .
A trifecta bet is one where the player tries to predict the top three finishers in order. How many possible trifecta bets are there for this race?
22 .
A superfecta bet is one where the player tries to predict the top four finishers in order. How many possible superfecta bets are there for this race?
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