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The counting techniques developed in Lessons 6.1 and 6.2 are only onemethod short of forming a fairly complete tool kit for modeling a varietyof probability problems. In this lesson, you consider a technique forcounting in situations in which the order of occurrence is unimportant.

CombinationsThe game proposed by Hilary in Lesson 6.1 is a simple lottery in whichparticipants select two numbers from nine printed on a card.Participation requires a selection, not an ordering. That is, if the winningnumbers are 2 and 6, it does not matter whether the participant selects 2or 6 first.

The term combination is used to describe a selection of severalobjects. The number of combinations in a situation can be counted bymodifying the technique used to count permutations. For example, ifHilary’s game requires an ordering of two numbers instead of a selection,then the number of ways of filling out a ticket is counted as a permutation: P(9, 2) = = 72. Because this permutation counts a

pair such as 2 and 6 as different from the pair 6 and 2, every possible pairis counted twice. Thus, the number of combinations of two things selectedfrom a group of 9 is 72/2 = 36. Two commonly used symbols for thenumber of combinations of two things from a group of nine are C(9, 2) or 9C2.

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Lesson 6.3

Counting Techniques,Part 2

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318 Chapter 6 • Counting and Probability

If Hilary’s game requires picking three numbers in the proper order,then the number of ways of filling out a card is P(9, 3) = 504. If order doesnot matter, then 504 is too large. For example, if the winning numbers are2, 5, and 8, then 504 counts any arrangement of 2, 5, and 8 as different.The number of ways of arranging 2, 5, and 8 is 3 × 2 × 1 = 6. Therefore, 504 is six times too large, and C(9, 3) = = 504/6 = 84.

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

. But, P(n, m) = so C(n, m) = .

Since there are 36 ways of filling in one of Hilary’slottery tickets, the probability that any one ticket wins is1/36, or about .028. If 1,000 tickets are sold, Hilary canexpect about 1,000 × .028 = 28 winners. If the gamerequires the selection of three numbers, the probability asingle ticket wins is 1/84, or about .012. If 1,000 tickets aresold, about 1,000 × .012 = 12 winners can be expected.

Using Combinations with Other Counting TechniquesCombinations are often used along with other countingtechniques. For example, the 17-member student councilat Central High consists of 9 girls and 8 boys. A committeeof 4 council members is being selected. If the committeemembers are not arranged in a particular way (i.e., chair,

secretary, and so forth), then the order of selection is unimportant. Thenumber of ways the committee can be selected is C(17, 4) = = 2,380.

If the committee must have two girls and two boys, there are C(9, 2) = = 36 ways of selecting the 2 girls and C(8, 2) = = 28 ways

of selecting the two boys. Because the committee must consist of 2 girlsand 2 boys, apply the multiplication principle to conclude that there are36 × 28 = 1,008 ways of forming the committee. If the 4 committeemembers are selected at random, the probability that the committeeconsists of 2 girls and 2 boys is , or about .424.

Now suppose that the committee must consist of either all boys or allgirls. There are C(9, 4) = = 126 ways of selecting 4 girls and C(8, 4) =

= 70 ways of selecting 4 boys. Because the committee must consist of

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Combinations can becalculated by using acalculator’s factorialfunction or, on somecalculators, by using acombination function.

Technology Note


319Lesson 6.3 • Counting Techniques, Part 2

either 4 girls or 4 boys and because all-boy and all-girl committees aremutually exclusive, apply the addition principle to conclude that thereare 126 + 70 = 196 ways of forming the committee. Again, if the 4committee members are selected at random, the probability thecommittee consists of either all boys or all girls is , or about .082.

Exercises1. Which is larger: C(10, 2) or C(10, 8)?

2. Find the sum of all possible combinations of four things. That is,find C(4, 0) + C(4, 1) + C(4, 2) + C(4, 3) + C(4, 4). Do the same forall possible combinations of three things and all possiblecombinations of five things. On the basis of your results, make aguess about the sum of all possible combinations of six things.Describe any pattern you notice.

3. In this lesson the number of all-boy four-person committees on the Central High student council is calculated as C(8, 4) = 70, thenumber of all-girl four-person committees is calculated as C(9, 4) = 126, and the number of four-person committees that arehalf boys and half girls is calculated as C(8, 2) × C(9, 2) = 1,008.

a. How many four-person committees consist of three girls and one boy?

b. How many committees consist of one girl and three boys?

c. Find the sum of the numbers of committees that consist of fourboys, no boys, two boys, three boys, and one boy. Compare thissum with the total number of four-person committees calculatedby C(17, 4) in this lesson (see page 318).

4. Darrell Dewey has just left his Central High social studies class andbumped into his friend Carla Cheetham. Darrell informs Carla thatMs. Howe gave a ten-question true/false quiz today. When Carlaasks about the quiz, Darrell says he found it easy and thinks thatfour of the answers are false.

a. When Carla takes the quiz, in how many ways can she selectfour questions to mark false?

b. In how many ways can Carla select six questions to mark true?

c. In how many ways can Carla fill in the quiz if she ignoresDarrell’s hint?

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5. A standard deck of cards contains 13 different cards from each offour suits: spades and clubs, which are black in color, anddiamonds and hearts, which are red in color.

a. In how many ways can 2 cards be dealt from a standard 52-card deck?

b. In how many ways can 2 red cards be dealt from a standard 52-card deck?

c. What is the probability that 2 cards dealt from a standard 52-card deck are both red?

6. Maria has a part-time summer job selling ice cream from a smallvehicle she drives through residential areas of her community. Shecarries eight different flavors and sells a two-scoop cone for $1.80.

a. How many two-scoop cones are possible if both scoops are thesame flavor?

b. How many two-scoop cones are possible if each scoop is adifferent flavor?

c. All together, how many two-scoop cones are possible?

7. Hedy Foans, who writes a music column in the Central HighScribbler, decides to poll students on their favorite songs. Sheprepares a list of ten current favorites, from which students areasked to rank their top three. In how many ways can a student picka first, second, and third choice from Hedy’s ten?



8. Ms. Howe has a planter in one of her classroom windows that isdivided into five sections. She purchases two geraniums and threemarigolds to plant in the five spaces.

a. In how many ways can Ms. Howe select the two sections inwhich to plant the geraniums?

b. In how many ways can Ms. Howe select the three sections inwhich to plant the marigolds?

9. In February 1992, an Australian company sent representatives toVirginia in an attempt to purchase one ticket for every possibleselection in the state’s lottery. The representatives spread theirpurchases among eight retail chains that had a total of 125 outlets.One representative bought a total of 2.4 million tickets at a singleretail chain headquarters. When time ran out, the group hadpurchased 5 million tickets, or about 70% of all possible selections.

One of the tickets purchased by the group matched the winningnumbers: 8, 11, 13, 15, 19, 20. After a controversy over the legalityof the purchase, the lottery decided to award the $27 millionjackpot to the Australian group, which represented about 2,500investors who paid an average of $3,000 each. Each investor stoodto receive an average of $10,800, at the rate of $540 a year over the20-year payment period.

a. At the time of the Australian purchase, a Virginia lottery ticketcontained the numbers 1 through 44, from which a participantselected six. In how many ways can a selection be made?

b. If it takes 5 seconds to purchase a Virginia lottery ticket, howlong would it take one person working 40 hours a week topurchase a ticket for every possible selection?

c. If each Virginia lottery form has space for five entries and if eachform has a thickness of 0.003 inch, how thick is a stack of formsof all possible selections?

d. Until October of 1999, a Florida lottery ticket contained thenumbers 1 through 49, from which a participant selected six. Inhow many ways can this be done?

e. An individual once bought 80,000 tickets in the Florida lottery.What was this person’s probability of winning a share of the $94 million jackpot that had accrued at that time?



f. After October of 1999, the Florida lottery required the selectionof six numbers from 53. How does the probability of winning thejackpot in the Florida lottery today compare with the probabilityof winning the jackpot prior to the change?

10. Most lotteries include several prizes besides the jackpot. Forexample, the Florida lottery gives second prizes to tickets thatmatch 5 of the 6 winning numbers, third prizes to those that match4 of the 6, and fourth prizes to those that match 3 of the 6.

a. How many different ways are there to receive a second prize?(Hint: The ticket must match 5 of the 6 winning numbers and 1of the 47 non-winning numbers.)

b. How many different ways are there to receive a third prize?

c. How many different ways are there to receive a fourth prize?

11. Chapter 1 discusses various voting models. Suppose there are sevenchoices on a ballot.

a. In how many ways can a voter rank the seven choices?

b. Recall that when approval voting is used, the choices are notranked. In how many ways can you select three choices of whichto approve?

12. Dee Noat, the director of CentralHigh’s music department, isholding tryouts for the school’sjazz band. There are 7 studentscompeting for three saxophonepositions, 8 for two piano spots, 5 for two percussion spots, and 12 for three places as guitarists. In how many ways can Deeselect her band?

Frank and Ernest © reprinted by permission of Newspaper Enterprise Association, Inc.



13. The figure below was drawn by marking nine equally spaced pointson the circumference of a circle and connecting every pair of points.

a. How many chords are there?

b. Number the points from 1 through 9, and explain why drawingthe chords is analogous to filling out every possible selection inHilary’s lottery.

c. Recall that a complete graph is one in which every pair ofvertices is connected with an edge. How many edges are there ina complete graph with ten vertices?

14. Emily’s Pizza Emporium can prepare a pizza with any one or moreof nine ingredients. In how many different ways can a pizza beordered at Emily’s? (Hint: A pizza can be ordered with oneingredient, or two ingredients, or three ingredients, . . . .)

15. College Inn Pizza claims that it offers 105 different two-toppingpizzas. How many different toppings do you think College Inn Pizzauses? Explain.

16. Carl Burns, coach of the Central High Lions basketball team, has 12players on his squad. Of these, 3 are centers, 4 are forwards, and 5are guards.

a. Is it correct to say that a team requires a center and 2 forwardsand 2 guards, or is it correct to say that a team requires a centeror 2 forwards or 2 guards?

b. In how many ways can Coach Burns select his starting team?



17. A telephone exchange consists of all seven-digit phone numberswith the same three-digit prefix.

a. How many different phone numbers are possible in a givenexchange?

b. If a community has 95,000 telephone subscribers, what is theminimum number of exchanges needed?

c. How many phone numbers are possible in a given three-digitarea code? (Assume all possible exchanges are permitted.)

18. Allison Gerber, a math teacher at Central High, gives prizes tostudents in her class who improve their average grade. At the endof each term she places the names of all qualifying students in acontainer and draws three.

a. If there are 21 qualifying students and the prizes are threeCentral High Lions T-shirts, in how many ways can the prizes be awarded?

b. If there are 21 qualifying students and the prizes are a newcalculator, a Lions T-shirt, and a discrete mathematics book, inhow many ways can the prizes be awarded?

19. Electronic data encryption is important to most people because theirbank accounts and other financial data are accessible over theInternet. For many years, the United States government certified a56-bit encryption system. Each bit can be either a 0 or a 1.

a. Read the news article on page 325. Explain how the number ofunique DES keys is calculated.

b. The article reports that in 2004, a key could be broken in 1/64thof the time it took to break it in 1998. Does this claim seemreasonable if the article’s claim that computer speeds doubleevery 18 months is accurate?

c. The AES encryption system uses 128 bits. Assuming thatcomputer speeds double every 18 months, in what year wouldyou expect a 128-bit key to be breakable in one week using asystem comparable to the one reported in the article?

d. Does the article’s use of the word “combination” seemappropriate to you? Explain.



20. Dominoes come in different-sized sets. A double-six set is the mostcommon. In a double-six set, each half of a domino may have anynumber of spots from 0 through 6. The two halves of a givendomino in the set pair a number of spots with the same number ofspots or a different number of spots.

a. How many dominoes with the same number of spots on eachhalf are there in a double-six set?

b. If every possible pairing is included in the set, how manydominoes with a different number of spots on each half arethere in a double-six set?

c. What is the total number of dominoes in a double-six set?

DES Encryption is Inadequate says NIST

Computer WeeklyJuly 30, 2004

The National Institute ofStandards and Technology(NIST) is proposing that theData Encryption Standard(DES) lose its certification foruse in software products sold tothe government.

The algorithm uses a 56-bit key to encrypt blocks of data,and can produce up to72,000,000,000,000,000 unique keys.

While that number of uniquecombinations was formidablein the 1970s and 1980s, giventhe power of computers at thattime, experts were aware thatthe growth of computing powerwould, in time, render thealgorithm breakable, and thatDES had at most a 15-year lifespan, according to NIST.

By the 1990s, computers hadbecome powerful enough thatbreaking the DES algorithmwas achievable, even forgroups with limited resources.In a 1998 experiment fundedby the non-profit civil libertiesgroup the Electronic FrontierFoundation, Paul Kocher andhis colleagues designed amachine for about $250,000that could break one DES key a week.

With computers doubling inspeed every 18 months, asimilar system designed with2004 technology couldpresumably break a key in1/64th of that time using so-called “brute force” methods,which essentially try everypossible key combination untilthe correct combination isguessed.



d. Write a description of the way a domino in a double-six set isformed. Explain how the words and or or in your descriptionreflect the calculations you made to obtain your answer to part c.

e. If you select a domino at random from a double-six set, what is the probability that it has the same number of spots on each half?

f. How many dominoes are there in a double-twelve set?

21. How many different sums of money can be made from a $1 bill, a$5 bill, a $10 bill, and a $20 bill? (Hint: You can use one bill at atime or two bills at a time or three bills at a time or four bills at a time.)

22. Many card games involve 5-card hands. (See the description of astandard deck of cards in Exercise 5.)

a. How many different 5-card hands can be dealt from a standard52-card deck?

b. In how many ways can a selection of 3 aces be made from the 4aces that are found in a standard deck?

c. In how many ways can 3 cards of the same kind (aces, twos,threes, and so forth) be dealt from a standard deck? (Hint: Youcan deal 3 aces or 3 twos or 3 threes or . . . .)

d. Repeat part c for 2 cards of the same kind.

e. In how many ways can a hand consisting of 3 of one kind and 2of another (a full house) be dealt from a standard deck?

23. To win the jackpot in the California Fantasy 5 lottery game, aparticipant must match 5 numbers from 39 that are available. Ifyou buy ten tickets per week in the California lottery, about howoften could you expect to win the jackpot? Explain.

24. Some bike locks allow the user to set a four-digit codethat opens the lock. These locks are convenientbecause the user can select a familiar number and isthereby less likely to forget the code.

a. How many different codes are possible with such a lock?

b. Locks like these are often called combination locks.Do you think this is an appropriate name? Explain.



25. The term odds is often used in the media. Although related, oddsare different from probabilities, and the difference can causeconfusion. The definition of the odds in favor of an event is theratio of the number of ways the event can occur to the number ofways it can fail. For example, consider a simple game in which youroll a single die and win if one or two spots show and loseotherwise. The odds in favor of your winning are 2:4 or 1:2. Theprobability of your winning is or . The odds against yourwinning are 4:2 or 2:1.

a. Based on the news article below about the 2013 Kentucky Derby,what is the probability that Revolutionary will win?

b. Revolutionary lost the race. According to the article, whatprobability was assigned to this event?

c. Odds are expressed in various ways. Colons and dashes arecommonly used. But odds can also be written as fractions andconverted to decimals. In the news article on the 2013 KentuckyDerby, use decimals to compare the odds in favor of horses withthe probability of their winning. When are the odds and theprobability nearly equal?

13

26

Kentucky Derby: Revolutionary Remains Pre-Race Favorite

SB NationMay 4, 2013

With post time now less than anhour and a half away, the odds forthe 2013 Kentucky Derby remainsteady. Revolutionary, who becamethe favorite as odds moved due towet conditions, is still favored at 5-1.

Goldencents remains at 7-1 odds, buthas now been joined in a tie forsecond place by Orb, whose oddsrecently moved from 8-1 to 7-1.

Normandy Invasion at 8-1,Itsmyluckyday at 9-1 and Verrazanoat 9-1 round out the top contenders.Falling Sky continues to have thelowest odds although the horse hasmoved up from 37-1 to 36-1. Vyjackremains with long odds at 29-1.



Projects26. Research one or more of the lotteries in your area. How large are

the jackpots? How many tickets are usually sold? What portion ofthe proceeds goes to the players? What happens to the rest of themoney? Are there any rules to prevent the kind of purchase made by the Australian group in the Virginia lottery(see Exercise 9)? What kinds of strategies are known to be used by players?

27. Investigate probabilities of common card hands. Show how tocalculate as many as possible.

Whist Players Astonished After Each Receives FullSuit in One Hand

The Daily MailNovember 24, 2011

It is an occurrence that comes withmind-boggling odds of a thousandquadrillion – or a thousand millionmillion million million – to one.

But a group of whist-playing pensionerssay they were stunned when eachplayer was dealt a complete suit in anopening hand.

Wenda Douthwaite, 77, and her threefriends were left ‘gobsmacked’ duringthe game in their village hall last week.

Mathematicians say the odds of thishappening are a jaw-dropping2,235,197,406,895,366, 368,301,559,999to one.

The 28-digit figure is the equivalentodds of a person finding a specific dropof water in the Pacific Ocean.

Mrs Douthwaite, from Kineton,Warkickshire, who has attended whist

drives for 50 years, said: ‘We’ve neverseen anything like it before. Everythingwas done as usual.

‘The cards were shuffled, cut and dealtas normal but that was the only thingthat was normal. And it was the firstgame of the night as well. As soon as Ipicked up my cards I saw I had acomplete set of spades.

‘Suddenly someone around the tablesaid they’d got a complete suit too. Wecompared cards and were totallyshocked when one of us had all thehearts, another had the diamonds,another had the clubs and I had thespades. I was shaking when we laid thecards down on the table.


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