Deck 7: Sets and Counting

Question 1

Evaluate $C ( 9,6 )$ .

A) 84
B) 6
C) 12
D) 504
E) 60,480

Answer 1

84

Question 2

The "Dogs of the Dow" refer to the stocks listed on the Dow with the highest dividend yield. The table shows the top ten stocks of the "Dogs of the Dow" list in February, 2005. You decide to make a small portfolio consisting of a collection of seven of the top ten Dogs of the Dow. How many portfolios are possible $$\begin{array} { | c | c | c | c | } \hline \text { Symbol } & \text { Company } & \text { Price } & \text { Yield } \\\hline \text { SBC } & \text { SBC } & 23.62 & 5.46 \% \\\hline \text { Communications } & \text { General Motors } & 36.64 & 5.46 \% \\\hline \text { MRK } & \text { Merck } & 28.02 & 5.42 \% \\\hline \text { MO } & \text { Altria } & 63.20 & 4.62 \% \\\hline \text { VZ } & \text { Verizon } & 35.68 & 4.32 \% \\\hline \text { JPM } & \text { JP Morgan Chase } & 37.00 & 3.68 \% \\\hline \text { C } & \text { Citigroup } & 48.38 & 3.64 \% \\\hline \text { PFE } & \text { Pfizer } & 24.35 & 3.12 \% \\\hline \text { DD } & \text { DuPont } & 47.23 & 2.96 \% \\\hline \text { GE } & \text { General Electric } & 35.75 & 2.46 \% \\\hline\end{array}$$

A) 5,040 portfolios
B) 21 portfolios
C) 24 portfolios
D) 120 portfolios
E) 56 portfolios

Answer 2

120 portfolios

Question 3

A bag contains 3 red marbles, 6 blue marbles, 2 yellow marbles, and 3 lavender marbles. How many sets of 4 marbles include at most one of the yellow marbles

A)990
B) 495
C) 715
D) 935
E) 1,430

Answer 3

935

Question 4

The number of ordered lists of 5 different items, chosen 8 at a time is 56.

Question 5

A bag contains 3 red marbles, 6 blue marbles, 2 yellow marbles, and 3 lavender marbles. How many sets of 3 marbles include none of the yellow marbles

A) 220
B) 2
C) 1,320
D) 660
E) 504

Question 6

A poker hand consists of five cards from a standard deck of 52. Find the number of different poker hands of four of a kind (all four of one denomination and one of another). ​

A)156 hands
B) 9,984 hands
C) 52 hands
D) 2,496 hands
E) 624 hands

Question 7

Evaluate $C ( 7,4 )$ .

A) 840
B) 6
C) 24
D) 210
E) 35

Question 8

Evaluate $P ( 8,5 )$ .

A) 6
B) 56
C) 6,720
D) 13
E) 40

Question 9

If 8 persons meet at a reunion and each person shakes hands exactly once with each of the others, what is the total number of handshakes ?

A)?56 handshakes
B) ?28 handshakes
C) ?64 handshakes
D) ?168 handshakes
E) ?16 handshakes

Question 10

How many ordered sequences are possible that contain 2 objects chosen from 9

A) 72
B) 5,040
C) 2
D) 725,760
E) 36

Question 11

The number of ordered lists of 7 different items, chosen 4 at a time is 840.

Question 12

How many ordered lists are there of 4 items chosen from 6 ?

A) 15
B) 2
C) 24
D) 48
E) 360

Question 13

A bag contains 3 red marbles, 6 blue marbles, 2 yellow marbles, and 1 lavender marble. How many possible sets of three marbles are ther

A) 220
B) 1,320
C) 6
D) 720
E) 440

Question 14

Find the indicated value. ​7!
​

A)​24
B) ​720
C) ​120
D) ​40,320
E) ​5,040

Question 15

How many eight-letter sequences are possible that use the letters f, i, t, t, i, n, g, s

A)720
B) 4
C) 20,160
D) 16
E) 10,080

Question 16

Evaluate $P ( 7,5 )$ .

A) 42
B) 21
C) 5,040
D) 120
E) 2,520

Question 17

In how many ways can a 6-card hand containing 4 kings and 2 queens be dealt from a deck of 36 cards

A)1
B) 288
C) 576
D) 6
E) 4

Question 18

A poker hand consists of five cards from a standard deck of 52. Find the number of different poker hands of the flush (five cards all of the same suit, but not consecutive denominations: A, 2, 3, 4, 5 up through 10, J, Q, K, A). ​

A)1,287 hands
B) 5,108 hands
C) 792 hands
D) 5,148 hands
E) 1,247 hands

Question 19

Evaluate $\frac { 8 ! } { 7 ! }$ .

A)56
B) 16
C) 8
D) 1
E) 15

Question 20

How many six-letter sequences are possible that use the letters g, o, b, l, i, n once each ?

A)216
B) 120
C) 720
D) 6
E) 36

Question 21

Evaluate .
​ __________

Question 22

Evaluate .
​ __________

Question 23

In how many ways can a 5-card hand containing 4 kings and 1 queen be dealt from a deck of 52 cards

Question 24

How many five-letter sequences are possible that use the letters b, l, a, c, k once each

Question 25

A bag contains 3 red marbles, 5 blue marbles, 3 yellow marbles, and 2 lavender marbles. How many sets of 3 marbles include none of the yellow marbles

Question 26

Professor Tough's final examination has 11 true/false questions followed by 5 multiple choice questions. In each of the multiple choice questions you must select the correct answer from a list of 4. How many answer sheets are possible ?

A)440
B) 1,280,000
C) 3,072
D) 40,960
E) 2,097,152

Question 27

Evaluate .
​ __________

Question 28

Rather than study for math, you and your buddies decide to get together for a marathon movie-watching popcorn-gorging event on Saturday night. You decide to watch four movies selected at random from the list. How many sets of four movies are possible $$\begin{array} { | c | c | } \hline \text { Name } & \text { Revenue (\$ million) } \\\hline \text { The Grudge } & \$ 9.24 \\\hline \text { Ray } & \$ 6.54 \\\hline \text { The Forgotten } & \$ 6.01 \\\hline \text { Friday Night Lights } & \$ 5.99 \\\hline \text { Cellular } & \$ 5.85 \\\hline \text { Shall We Dance } & \$ 5.82 \\\hline \text { The Village } & \$ 5.58 \\\hline \text { AVP: Alien Vs. Predator } & \$ 5.46 \\\hline \text { Catwoman } & \$ 5.10 \\\hline \text { Sky Captain and the World of } \\\text { Tomorrow } & \$ 5.09 \\\hline\end{array}$$

A) 5,040 sets
B) 210 sets
C) 720 sets
D) 182 sets
E) 91 sets

Question 29

Evaluate .
​ __________

Question 30

A computer byte consists of 8 bits, each bit being either a 0 or a 1. If characters are represented using a code that uses 2 bytes for each character, how many characters can be represented ?

A)128
B) 100
C) 65,536
D) 512
E) 32

Question 31

The "Dogs of the Dow" refer to the stocks listed on the Dow with the highest dividend yield. The table shows the top ten stocks of the "Dogs of the Dow" list in February, 2005. You decide to make a small portfolio consisting of a collection of six of the top ten Dogs of the Dow. $$\begin{array} { | c | c | c | c | } \hline \text { Symbol } & \text { Company } & \text { Price } & \text { Yield } \\\hline \text { SBC } & \begin{array} { c } \text { SBC } \\\text { Communications }\end{array} & 23.62 & 5.46 \% \\\hline \text { GM } & \text { General Motors } & 36.64 & 5.46 \% \\\hline \text { MRK } & \text { Merck } & 28.02 & 5.42 \% \\\hline \text { MO } & \text { Altria } & 63.20 & 4.62 \% \\\hline \text { VZ } & \text { Verizon } & 35.68 & 4.32 \% \\\hline \text { JPM } & \text { JP Morgan Chase } & 37.00 & 3.68 \% \\\hline \text { C } & \text { Citigroup } & 48.38 & 3.64 \% \\\hline \text { PFE } & \text { Pfizer } & 24.35 & 3.12 \% \\\hline \text { DD } & \text { DuPont } & 47.23 & 2.96 \% \\\hline \text { GE } & \text { General Electric } & 35.75 & 2.46 \% \\\hline\end{array}$$
How many portfolios are possible ?
__________ portfolios
How many of these portfolios contain SBC but not MO ?
__________ portfolios
How many of these portfolios contain at most one stock priced above $40 ?
__________ portfolios

Question 32

A poker hand consists of five cards from a standard deck of 52. Find the number of different poker hands of the flush (five cards all of the same suit, but not consecutive denominations: A, 2, 3, 4, 5 up through 10, J, Q, K, A).
​
__________ hands

Question 33

A bag contains 5 red marbles, 4 blue marbles, 2 yellow marbles, and 2 lavender marbles. How many possible sets of three marbles are there

Question 34

A bag contains 3 red marbles, 6 blue marbles, 2 yellow marbles, and 2 lavender marbles. How many sets of 4 marbles include at most one of the yellow marbles

Question 35

How many eight-letter sequences are possible that use the letters p, r, o, p, e, r, t, y

Question 36

A poker hand consists of five cards from a standard deck of 52. Find the number of different poker hands of two of a kind (two of one denomination and three of different denominations).
​
__________ hands

Question 37

Rather than study for math, you and your buddies decide to get together for a marathon movie-watching popcorn-gorging event on Saturday night. You decide to watch six movies selected at random from the list.
?? $$\begin{array} { | c | c | } \hline \text { Name } & \text { Revenue ( } \text { ( million) } \\\hline \text { The Grudge } & \$ 9.24 \\\hline \text { Ray } & \$ 6.54 \\\hline \text { The Forgotten } & \$ 6.01 \\\hline \text { Friday Night Lights } & \$ 5.99 \\\hline \text { Cellular } & \$ 5.85 \\\hline \text { Shall We Dance } & \$ 5.82 \\\hline \text { The Village } & \$ 5.58 \\\hline \text { AVP: Alien Vs. Predator } & \$ 5.46 \\\hline \text { Catwoman } & \$ 5.10 \\\hline \begin{array} { c } \text { Sky Captain and the World of } \\\text { Tomorrow }\end{array} & \$ 5.09 \\\hline\end{array}$$
a. How many sets of six movies are possible ?
__________ sets
b. Your best friends, the Lara twins, refuse to see "Shall We Dance" on the grounds that it is "for girliemen" and also insists that at least one of "Alien Vs. Predator" or "Sky Captain" be among the movies selected. How many of the possible groups of six will satisfy the twins ?
__________ groups
c. Comparing the answers in parts (a) and (b), would you say the Lara twins are more likely than not to be satisfied with your random selection ?
__________ (yes or no )

Question 38

Evaluate .
​ __________

Question 39

Find the indicated value.
​
7!

Question 40

If 10 persons meet at a reunion and each person shakes hands exactly once with each of the others, what is the total number of handshakes ?
__________ handshakes

Question 41

A Social Security number is a sequence of nine digits. How many Social Security numbers are possible that have the same first 5 digits ?

A)10,000 numbers
B) 40 numbers
C) 50 numbers
D) 1,048,576 numbers
E) 100,000 numbers

Question 42

Find how many outcomes are possible for the decision algorithms. $\begin{array}{ll}{\text { Step } 1} & \text { Step } 2 \\\text { Alternative 1: } 1 \text { outcome } & \text { Alternative } 1: 1 \text { outcome } \\\text { Alternative } 2: 1 \text { outcome } & \text { Alternative } 2: 1 \text { outcome } \\\text { Alternative } 3: 2 \text { outcomes } &\end{array}$

A)6
B) 8
C) 3
D) 4
E) 2

Question 43

Professor Oger must choose between an extra writing assignment and an extra reading assignment for the upcoming spring break. For the writing assignment, there are 4 essay topics to choose from and three different mandatory lengths (30 pages, 35 pages, or 40 pages). The reading topic would consist of one scholarly biography combined with one volume of essays. There are 5 biographies and two volumes of essays to choose from. How many choices does she have ?

A)49 choices
B) 84 choices
C) 19 choices
D) 22 choices
E) 70 choices

Question 44

An experiment requires a sequence of three steps. The first step can result in 5 possible outcomes, the second in 2 possible outcomes, and the third in 3 possible outcomes. What is the total number of outcomes possible

Question 45

An experiment requires a sequence of three steps. The first step can result in 6 possible outcomes, the second in 3 possible outcomes, and the third in 5 possible outcomes. What is the total number of outcomes possible ?

A)14
B) 180
C) 54
D) 90
E) 75

Question 46

A test requires that you answer first Part A and then either Part B or Part C. Part A consists of 7 true/false questions, Part B consists of 9 multiple choice questions with one correct answer out of 5, and Part C requires you to match 8 questions with 8 different answers. (Each question matches exactly one correct answer.) How many different completed answer sheets are possible ?

A) 79,000,000,000
B) 1,993,573
C) 78,755,160,960
D) 255,160,960
E) 307,257,753,600

Question 47

How many three-letter sequences are possible that contain only the letters R and D, with D occurring only once ?

A)4 sequences
B) 27 sequences
C) 9 sequences
D) 6 sequences
E) 3 sequences

Question 48

A test requires that you answer either Part A or Part B. Part A consists of 16 true-false questions and part B consists of 10 multiple-choice questions with one correct answer out of 3. How many different completed answer sheets are possible ?

A)65,566 sheets
B) 1,966,080 sheets
C) 59,081 sheets
D) 960 sheets
E) 124,585 sheets

Question 49

In an eight-cylinder engine, the even-numbered cylinders are on the left and the odd-numbered cylinders are on the right. A good firing order is a sequence of the numbers 1 through 8 in which right and left sides alternate. How many possible good firing sequences are there ?

A)40,320 sequences
B) 512 sequences
C) 576 sequences
D) 1,152 sequences
E) 24 sequences

Question 50

Some written languages, such as Chinese and Japanese, use tens of thousands of different characters. If a language uses 4,096 characters, a computer code for this language would have to use how many bits (1 or 0) per character ?

A)2 bits
B) 12 bits
C) 24 bits
D) 2,048 bits
E) 144 bits

Question 51

The World Almanac features a "perpetual calendar", a collection of possible calendars for every conceivable year. Find the lowest number of the calendars. ​

A)10 calendars
B) 28 calendars
C) 21 calendars
D) 14 calendars
E) 30 calendars

Question 52

Find how many outcomes are possible for the decision algorithms. $\begin{array}{cc}\text { Alternative 1 } & \text { Alternative 2 } \\\text { Step 1: six outcomes } & \text { Step 1: three outcomes } \\\text { Step 2: two outcomes } & \text { Step 2: two outcomes } \\& \text { Step 3: two outcomes }\end{array}$

A)56
B) 15
C) 24
D) 40
E) 26

Question 53

A surgical procedure requires choosing among three alternative methodologies. The first methodology can result in 5 possible outcomes, the second in 4 possible outcomes, and the third in 3 possible outcomes. What is the total number of outcomes possible

Question 54

How many possible calendars are there that have February 11 falling on a Monday, or Tuesday ?

A)2 calendars
B) 5 calendars
C) 4 calendars
D) 1 calendars
E) 7 calendars

Question 55

A telephone number consists of a sequence of seven digits not starting with a 0 or a 1. How many phone numbers begin with either 7,342, 7,423, or 7,224 ?

A)2,187 numbers
B) 27,000 numbers
C) 1,000 numbers
D) 3,000 numbers
E) 90 numbers

Question 56

Find how many outcomes are possible for the decision algorithms. $\begin{array}{cl}{\text { Step } 1} & {\text { Step } 2} \\\text { Alternative 1:1 outcome } & \text { Alternative 1:1 outcome } \\\text { Alternative 2: } 1 \text { outcome } & \text { Alternative } 2: 1 \text { outcome } \\& \text { Alternative } 3: 2 \text { outcomes }\end{array}$

A)8
B) 6
C) 3
D) 2
E) 4

Question 57

A surgical procedure requires choosing among three alternative methodologies. The first methodology can result in 7 possible outcomes, the second in 6 possible outcomes, and the third in 2 possible outcomes. What is the total number of outcomes possible ?

A)9
B) 8
C) 15
D) 13
E) 84

Question 58

Dripping wet after your shower, you have clean forgotten the combination of your lock. It is one of those "standard" combination locks, which uses a three number combination with each number in the range 0 through 39. All you remember is that the second number is either 25 or 35, while the third number ends with a 4. In desperation, you decide to go through all possible combinations using the information you remember. Assuming that it takes about 10 seconds to try each combination, what is the longest possible time you may have to stand dripping in front of your locker
Assume that 4 does not end with a 4 .

A)240 minutes
B) 39 minutes
C) 53 minutes
D) 8 minutes
E) 40 minutes

Question 59

While selecting candy for students in his class, Professor Murphy must choose between gummy candy and licorice nibs. Gummy candy packets come in 3 sizes, and packets of licorice nibs come in 3 sizes. If he chooses gummy candy, he must select gummy kangaroos, gummy lions, gummy bears, gummy worms or gummy dinos. If he chooses licorice nibs, he must choose between red and black. How many choices does he have ?

A)14
B) 17
C) 19
D) 21
E) 24

Question 60

A surgical procedure requires four steps. The first step can result in 3 possible outcomes, the second in 5 possible outcomes, and the remaining two can each result in 4 possible outcomes. What is the total number of outcomes possible ?

A)300
B) 240
C) 16
D) 60
E) 180

Question 61

A telephone number consists of a sequence of seven digits not starting with a 0 or a 1. How many phone numbers begin with either 764, 768, or 788

Question 62

The World Almanac features a "perpetual calendar", a collection of 14 possible calendars. Why does this suffice to be sure there is a calendar for every conceivable year

Question 63

A test requires that you answer either Part A or Part B. Part A consists of 21 true/false questions and part B consists of 9 multiple choice questions with one correct answer out of 5. How many different completed answer sheets are possible

Question 64

Professor Tough's final examination has 13 true-false questions followed by 5 multiple-choice questions. In each of the multiple choice questions you must select the correct answer from a list of 5. How many answer sheets are possible

__________ sheets

Question 65

While selecting candy for students in his class, Professor Murphy must choose between gummy candy and licorice nibs. Gummy candy packets come in 4 sizes, and packets of licorice nibs come in 2 sizes.
If he chooses gummy candy, he must select gummy lions, gummy bears, gummy worms or gummy dinos. If he chooses licorice nibs, he must choose between red and black. How many choices does he have

Question 66

Use the given information to complete the solution to the partially solved Venn diagram. $$n ( A ) = 24$$ , $n ( B ) = 19$ , $n ( C ) = 30$ , $n ( A \cap C ) = 6$ , and $n ( S ) = 51$

A)
B)
C)
D)
E)

Question 67

A test requires that you answer first Part A and then either Part B or Part C. Part A consists of 6 true/false questions, Part B consists of 9 multiple choice questions with one correct answer out of 5, and Part C requires you to match 8 questions with 8 different answers. (Each question matches exactly one correct answer.) How many different completed answer sheets are possible ?
__________ sheets

Question 68

Find how many outcomes are possible for the decision algorithms. ​​
__________ outcomes

Question 69

Find how many outcomes are possible for the decision algorithms. ​
__________ outcomes

Question 70

Professor Oger must choose between an extra writing assignment and an extra reading assignment for the upcoming spring break. For the writing assignment, there are 4 essay topics to choose from and three different mandatory lengths (30 pages, 35 pages, or 40 pages). The reading topic would consist of one scholarly biography combined with one volume of essays. There are 4 biographies and four volumes of essays to choose from. How many choices does she have

Question 71

A surgical procedure requires four steps. The first step can result in 6 possible outcomes, the second in 4 possible outcomes, and the remaining two can each result in 3 possible outcomes. What is the total number of outcomes possible

Question 72

A Social Security number is a sequence of nine digits. How many Social Security numbers are possible that have the same first 6 digits

Question 73

Find how many outcomes are possible for the decision algorithms.

Question 74

In an eight-cylinder engine, the even-numbered cylinders are on the left and the odd-numbered cylinders are on the right. A good firing order is a sequence of the numbers 1 through 8 in which right and left sides alternate.

a. How many possible good firing sequences are there ?
__________ sequences

b. How many good firing sequences are there that start with a cylinder on the left ?
__________ sequences

Question 75

Of the 5,700 students at medium Suburban College, 50 play collegiate soccer, 60 play collegiate lacrosse, and 80 play collegiate football. Only 4 play both collegiate soccer and lacrosse, 6 play collegiate soccer and football, and 15 play collegiate lacrosse and football. No students play all three sports. ​
Complete the following sentence: ______% of the college soccer players also play one of the other two sports at the collegiate level.
​

A)30
B) 36
C) 18
D) 22
E) 20

Question 76

A computer "byte" consists of 8 "bits", each bit being either a 0 or a 1. If characters are represented using a code that uses 4 bytes for each character, how many characters can be represented ?
__________ characters

Question 77

In December 2001 a search using the Web search engine Google^TM for "Costenoble" yielded 10,000 Web sites containing that word. A search for "Waner" yielded 67,000 sites. A search for sites containing both words yielded 4,393 sites. How many Web sites contained either "Costenoble" or "Waner" or both ?

A)14,393
B) 72,607
C) 81,393
D) 77,000
E) 71,393

Question 78

How many possible calendars are there that have February 15 falling on a Monday, or Tuesday ?
__________ calendars

Question 79

Some written languages, such as Chinese and Japanese, use tens of thousands of different characters. If a language uses 16,384 characters, a computer code for this language would have to use how many bits (1 or 0) per character

Question 80

Dripping wet after your shower, you have clean forgotten the combination of your lock. It is one of those "standard" combination locks, which uses a three-number combination with each number in the range 0 through 39. All you remember is that the second number is either 27 or 37, while the third number ends in a 4. In desperation, you decide to go through all possible combinations using the information you remember. Assuming that it takes about 10 seconds to try each combination, what is the longest possible time you may have to stand dripping in front of your locker ?
Assume that 4 does not end with a 4 .

__________ minutes