Lesson 6 - Weeblygvmath.weebly.com/.../chapter.06.02.counting.techniquest.part.i.pdf · Lesson 6.2 • Counting Techniques, Part 1 309 4. (See Exercise 8 in Lesson 6.1, page 303.)

In Lesson 6.1 you examined several situations in which the answer to thequestion, “In how many ways can this be done?” is important. Two techniquesthat can be used to answer this question are the multiplication principle andmaking a complete list. The latter can often be done with the aid of a treediagram or some other systematic procedure such as an algorithm.

In this and the next lesson, you will consider other countingtechniques, beginning with the addition principle.

The Addition PrincipleRecall that the multiplication principle says that if events A and B can occurin a and b ways, respectively, then events A and B can occur together in a × b ways. The addition principle says that if A and B can occur in a andb ways, respectively, then either event A or event B can occur in a + b ways.

For example, if the student council at Central High consists of 17members, of which 9 are girls and 8 are boys, and if one girl and one boyare to be selected to hold two different offices on the council, then thereare 9 × 8 = 72 ways of filling the two offices. If a single student, who maybe either a boy or a girl, is to be selected to hold a single office, then thereare 9 + 8 = 17 ways of making the selection.

The word and in the description of an event oftenindicates that the multiplication principle should beused, and the word or often indicates that theaddition principle should be used.

Lesson 6.2

Counting Techniques,Part 1

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306 Chapter 6 • Counting and Probability

The events “selecting a boy” and “selecting a girl” are calledmutually exclusive or disjoint because a person cannot be both a boyand a girl. On the other hand, events such as “selecting a member of yourschool’s football team” and “selecting a member of your school’sbasketball team” are not mutually exclusive if there is a person who is amember of both teams. When events are not mutually exclusive, theaddition principle requires a modification that you will consider in thislesson’s exercises.

Using the Multiplication and Addition Principles TogetherThe multiplication and addition principles are often used together, as thefollowing example shows. The Central High council members areconsidering a contest in which words of any length are made from theteam name Lions.

A word of one letter may be composed in only five ways: l, i, o, n, ands. A word of two letters requires a first letter and a second letter, whichmust be different from the first letter. Thus, there are 5 × 4 = 20 ways ofcomposing a word of two letters. A word of three letters requires a firstletter and a second letter and a third letter, so there are 5 × 4 × 3 = 60 waysof composing a word of three letters. Similarly, there are 5 × 4 × 3 ×

2 = 120 ways of composing a word of four letters and 5 × 4 × 3 × 2 × 1 =120 ways of composing a word of five letters.

A word may be composed by using oneletter or by using two letters or by using threeletters or by using four letters or by using fiveletters. Therefore, the total number of words is5 + 20 + 60 + 120 + 120 = 325. (Note that theevents “composing a word by using one letter,”“composing a word by using two letters,”“composing a word by using three letters,” and“composing a word by using four letters” aremutually exclusive.)


307Lesson 6.2 • Counting Techniques, Part 1

Factorials, Permutations, and ProbabilityThe calculation of the number of words of five lettersthat can be made from the letters of Lions requiresmultiplying all integers from 1 through 5. This productis an extension of the multiplication principle and isknown as the factorial of 5, or just 5 factorial. Afactorial is symbolized by an exclamation mark: 5!.Most calculators have a factorial key or function. If youhave never done a factorial on your calculator, trydoing so now.

The term permutation is often used to describe anordering (or arrangement) of several objects. Forexample, the game proposed by Pierre is one in whichorder matters: the words is and si are not the same.(Situations in which order does not matter areconsidered in Lesson 6.3.)

The number of permutations in a situation can becomputed by using your calculator’s factorial key. Forexample, to find the number of “words” of three lettersthat can be formed from the letters of lions, divide 5! by2!. Note that this calculation produces the correct resultbecause

5 × 4 × 3 = .

There are two commonly used symbols for the number ofpermutations of three things from a group of five: P(5, 3) or 5P3. Either iscalculated by evaluating the expression 5!/(5 – 3)!. (Many calculatorshave a special permutation function.)

In general, P(n, m) is calculated by evaluating the

expression

Lesson 6.1 began with several questions about the frequency withwhich certain events can occur. An event’s probability is the ratio of thenumber of ways the event can occur to the total number of possibilities inthat situation. For example, there are 325 “words” that can be formedfrom the letters of Lions, but only 20 of them have two letters. Thus, theprobability of forming a two-letter “word” from the letters of Lions is20/325. Probabilities can be expressed as fractions, decimals, or

n!(n – m)!

5 4 3 2 12 1

× × × ×

×

A factorial calculation on agraphing calculator.

Two ways to calculate apermutation on a graphingcalculator.

Technology Note



percentages. As a decimal, the probability of forming a two-letter“word” from Lions is .0615, so you would expect a two-letter word about6 times out of 100.

Because the numerator of a probability is never smaller than 0 andnever larger than the denominator, probabilities always range between 0and 1, inclusive. An event that cannot happen has probability 0; an eventthat is certain to happen has probability 1.

You now have several counting techniques at your disposal:

1. Making a list of all possibilities, for which tree diagrams are oftenhelpful

2. The multiplication principle and the related factorial andpermutation formulas

3. The addition principle

Skill at using these techniques develops with practice. The followingexercises help develop that skill and also demonstrate some refinementsof the three techniques.

Exercises1. Which is equivalent to P(10, 4), 10!/4! or 10!/6!? Find the value of

P(10, 4).

2. At right are the final USA Today 2012–2013 season rankings of highschool girls basketball teams. If you are a sportswriter voting for thetop teams and you can rank only your top 5, in how many ways canyou form your ranking from the 25 teams shown?

3. A multi-speed bicycle has a chain that ismoved to change the bicycle’s speed. Therider uses the bicycle’s front and rear shiftmechanisms to move the chain from onefront or rear sprocket to another.

a. If a bicycle has three front sprocketsand five rear sprockets, how manyspeeds does it have?

b. Is it correct to say that a particular speed requires a particularfront sprocket and a particular rear sprocket, or is it correct tosay that a particular speed requires a particular front sprocket ora particular rear sprocket?



4. (See Exercise 8 in Lesson 6.1, page 303.)

a. How many different “words” of any length can be made fromthe letters of insulate? (Hint: You can make a word of one letter ora word of two letters or a word of three letters . . . or a word ofeight letters.)

b. A group of students is considering entering the contest byprogramming a computer to print all possible ”words” that canbe made from the letters of insulate and then checking the listagainst an unabridged dictionary. If the computer prints thewords in four columns of 50 words each on a page of paper, howmany pages are needed?

Super 25 Girls Basketball Rankings

USA TodayApril 9, 2013

1. Duncanville High School, Duncanville TX

2. St. Mary's High School, Phoenix AZ

3. Riverdale High School, Murfreesboro TN

4. Marion County High School, Lebanon KY

5. Dutch Fork High School, Irmo, SC

6. St. John's College High School, Washington DC

7. Bishop O Dowd High School, Oakland CA

8. Hopkins High School, Hopkins MN

9. Malcom X. Shabazz High School, Newark NJ

10. Windward School, Los Angeles CA

11. Incarnate Word Academy, St. Louis MO

12. Mater Del High School, Santa Ana CA

13. Southwood High School, Shreveport LA

14. Millbrook High School, Raleigh NC

15. Dr. Phillips High School, Orlando FL

16. Hoover High School, Birmingham AL

17. St. Mary's High School, Stockton CA

18. Marian Catholic High School, Chicago Heights IL

19. Bedford North Lawrence High School, Bedford IN

20. Norcross High School, Norcross GA

21. Fairmont High School, Kettering OH

22. Spring-Ford Senior High School, Royersford PA

23. North Point High School, Waldorf MD

24. Grand Haven High School, Grand Haven MI

25. Long Island Lutheran High School, Glen Head NY



5. Some states have vehicle license “numbers” with three lettersfollowed by three digits. Often the letters I, O, and Z are not usedbecause they can be confused with the numerals 1, 0, and 2,respectively.

a. If these restrictions apply and if characters may be repeated,how many different license plates are possible?

b. What is the probability that a vehicle selected at random has alicense number that begins with CAT?

6. a. In how many ways can the coach of a baseball team arrange thebatting order of nine starting players?

b. A sportscaster once suggested that a baseball team try everypossible batting order for its nine starters in order to determinewhich one worked best. If a team decides to do so and plays onegame each day of the week with a different batting order ineach game, how long will it take to complete the experiment?

7. Three math students and three science students are taking finalexams. They must be seated at six desks so that no two mathstudents are next to each other and no two science students arenext to each other.

a. In how many ways can the students be seated if the desks are ina single row? (Hint: Draw six blanks and use the multiplicationprinciple.)

b. What is the probability that a math student occupies the firstseat in the row?

c. What is the probability that math students occupy the first seatand the last seat?

d. What is the probability that a math student occupies either thefirst seat or the last seat?

8. The multiplication principle states that the number of permutationsof the letters of the word math is 4!. The permutation formula says that P(4, 4) is . The denominator of this expression is 0!, 4

4 4!

( )!−which is meaningless. However, for 4! and to give the sameresult, what value must 0! have? Explain.

44 4

!( )!−



9. The U.S. Postal Service began using five-digit zip codes in 1963.Every post office was given its own zip code, which ranged from00601 in Adjuntas, Puerto Rico, to 99950 in Ketchikan, Alaska.

a. If the only five-digit zip code that could not be used was 00000,how many zip codes were possible in 1963?

b. Some five-digit zip codes are prone to errors because they arestill legal five-digit zip codes when read upside down. When thishappens, a letter goes to the wrong post office and must bereturned. How many zip codes are legal when read upsidedown? (Hint: Draw five blanks, think carefully about whichdigits can go in each blank, and apply the multiplicationprinciple.)

c. How many of the zip codes you counted in part b are not proneto errors because they read the same when turned upside down?

10. a. In how many ways can a person draw two cards from a standard52-card deck if the first card is returned to the deck before thesecond card is drawn?

b. In how many ways can two cards be drawn if the first card is notreturned?

11. The addition principle cannot be used as stated in this lesson whentwo events are not mutually exclusive. For example, if there are 15people on your school’s basketball team and 40 people on yourschool’s football team, then there are 55 ways of choosing oneperson from either team only if there are no people on both teams.

a. If there are eight people who play both football and basketball,in how many ways can a person be selected from either team?

b. Write the appropriate number of people in each of the threeregions of the following Venn diagram.

Football Basketball

? ? ?



c. Describe how the addition principle is applied when two eventsare mutually exclusive and how it is applied when two eventsare not mutually exclusive.

d. Event A and event B can occur in a and b ways, respectively, andevents A and B have c items in common. Write an algebraicexpression for the number of ways in which event A or event Bcan occur.

e. Central High’s soccer team has 37 members, and its basketballteam has 14 members. If there are a total of 43 studentsinvolved, how many are on both teams? Explain.

12. Before the 1992 major league baseball season began, Joe Torre,who then managed the St. Louis Cardinals, said he had picked hisstarting lineup. He also said he had determined his first threebatters but not the order in which they would bat. In how manyways could Joe arrange his batting order if the pitcher bats last?(Hint: Draw nine blanks and apply the multiplication principle.)

13. a. In how many different ways can a teacher arrange 30 students ina classroom with 30 desks?

b. The radius of the earth is approximately 6,370 kilometers. Astandard medical drop is cubic centimeter. Use the formula

earth in drops of water. Compare this with the number ofseating arrangements in part a.

14. There are three highways from Claremont to Upland and twohighways from Upland to Pasadena.

a. In how many ways can a driver select a route from Claremont toPasadena?

b. Is it correct to say that a trip from Claremont to Pasadenarequires a road from Claremont to Upland and a road fromUpland to Pasadena, or is it correct to say that the trip requires aroad from Claremont to Upland or a road from Upland toPasadena?

110

for the volume of a sphere, V = πr3, to find the volume of the 43

Claremont

Upland

Pasadena



c. In how many ways can a driver plan a round-trip fromClaremont to Upland and back?

15. Radio station call letters in the United States consist of three or fourletters, of which the first must be either a K or a W. Assuming thatletters may be repeated, determine the number of radio stationsthat can be assigned call letters.

The logo for radio station KURE at Iowa State University.

16. Six different prizes are given by drawing names from the 68 CentralHigh orchestra members attending the orchestra’s annual picnic. Inhow many ways can the prizes be given if no one can receive morethan one prize?

17. Three-digit telephone area codes were introduced in 1947. At thattime, the first digit could not be a 0 or a 1, the second digit could beonly a 0 or a 1, and the third could be any digit except 0.

a. How many area codes were possible?

b. Because of a shortage of area codes, beginning in 1995 any digitbecame a legal second digit. How many area codes were possibleafter 1995?

The original area codes.

604 403 306206

503

916

415

213

702

208

80

307

701

605

303

505602

402913316

405

915 214713

512

613

715

414

704

803901

502

514802218

712515319

816314

501

504

601 205 404

305

815217

312

6

204

406

1703304

612

618

7

317

812

61

51651

313

14 71715

416

51831

513

2 6

812

419

614

6

2

5

1

4

71

NoneNone



18. The UPC bar codes consist of two groupsof five digits each. One group, asassigned by the Uniform Code Council inDayton, Ohio, represents themanufacturer, and the other grouprepresents the products of thatmanufacturer.

a. How many different manufacturers can be encoded?

b. How many products can each manufacturer encode?

19. Various word puzzles with cash prizes can be found in places suchas Internet sites and newspapers. One type of puzzle is a modifiedcrossword in which a clue is given and only two choices are offered.

a. Consider such a puzzle with 20 questions, each having twopossible answers. How many different entries are possible? (Hint: Imagine 20 blanks and use the multiplication principle.)

Las Vegas Review-JournalNovember 16, 2012

The 411 is that 702 is about to get cozywith 725. And that means we're all goingto have to dial three more digits whenmaking local calls starting in 05 of 14.

In conventional speak, new residents ofSouthern Nevada, from Indian Springs in

the northwest toMesquite in thenortheast toLaughlin at the verybottom of the state,will be issued a 725area code in an

overlaying setup with the existing 702area code, beginning in June 2014.

The Public Utilities Commission of Nevadaon Thursday approved a second area codebecause available prefixes in the existing702 area code will be exhausted in 2014.

Because an overlaying area code willmean a new next-door neighbor will havea 725 area code while yours remains 702,local calls throughout the valley willrequire using an area code.

Overlaying area codes are becoming morecommon across the United States. Othercities about to implement overlaying areacodes are Boston and San Jose.

New Area Code in Southern Nevada Means 10-digit Calls



b. Someone embarks on the ambitious project of submitting everypossible entry. Suppose that it takes 5 minutes to do one entryand that the piece of paper on which an entry is written is 0.003inch thick. How long would it take to prepare the entries, andhow thick a stack of paper would result?

20. A carnival game called Chuk-a-Luk is similar to the one proposedby Chuck in Lesson 6.1, except that three dice are used.

a. In how many ways can three dice fall? Explain.

b. Determine the number of ways you can win $1, win $2, win $3,or lose $1 in the game of Chuk-a-Luk if you win $1 for each dieshowing your number. (Hint: You win $2 if the first and seconddice show your number and the third die doesn’t, or if the firstand third dice show your number and the second die doesn’t, orif the second and third dice show your number and the first diedoesn’t. Draw several sets of three blanks and then use themultiplication and addition principles.)

c. In the long run, do you think a player should expect to win orlose money in the game of Chuk-a-Luk? Explain.

21. The news article on page 300 discusses two types of bets that can bemade by selecting three digits from 0 through 9, “straight” bets and“box” bets. Players who made a box bet won with 911, 191, or 119.Discuss the difference in the two types of bets and determine theprobability of winning for each type.

22. Factorials can be described recursively. Let f(n) represent n!. Write arecurrence relation that expresses the relationship between f(n) andf(n – 1).

Projects23. Research the history of the study of probability. How did it begin?

What roles did Jerome Cardan, Blaise Pascal, and Pierre Fermatplay? What problems interested them?

24. Investigate the number of permutations of several objects, of whichsome look alike. For example, the letters of math can be arranged in4! = 24 ways; how many different permutations are there of theletters of look? What if there are several sets of identical letters, suchas in Mississippi? Write a summary that includes a general principlefor handling such situations and several examples.



25. Investigate the number of permutations of several objects arrangedin circular fashion. For example, Ann, Sean, Juanita, and Herb canbe seated along one side of a rectangular table in 4! = 24 ways. Inhow many different ways can they be seated around a circulartable? Write a summary that includes a general principle forhandling such situations and several examples. Explain yourinterpretation of the meaning of the word different.

26. Investigate the use of the addition principle with three events thatare not mutually exclusive. Suppose, for example, that the football,basketball, and track teams of Central High have 41, 15, and 34members, respectively. If 6 people play both football andbasketball, 7 are on both the basketball and track teams, 15 are onboth the football and track teams, and 4 play all three sports, howmany people are involved in one sport or another? Develop ageneral principle for modeling situations of this type and draw aVenn diagram to represent it. Can the principle be extended to fouror more events? How?


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Lesson 6 - Weeblygvmath.weebly.com/.../chapter.06.02.counting.techniquest.part.i.pdf · Lesson 6.2 • Counting Techniques, Part 1 309 4. (See Exercise 8 in Lesson 6.1, page 303.)