Deck 10: Counting Methods

Question 1

A male president and three representatives

A)ABCD, ACDE, CABD, CABE, EABC, EABD; 6
B)ABCD, CABD, CABE, EABD, ADCB, AEDC, CDBA, ECBA, EDBA; 9
C)ABCD, CABD, EABC; 3
D)ABCD, ACDE, ABCE, ABDE, CABD, CABE, CBDE, CADE, EABC, EABD, EBCD, EACD; 12

Answer 1

D

Question 2

Using the 36 possibilities found in the product table for rolling two dice, list and count the outcomes for which the sum
(for both dice)is the following.
Equal to 8

A)(2,6), (3,5), (4,4), (4,4), (5,3), (6,2); 6
B)(2,6), (3,5); 2
C)(2,6), (3,5), (4,4); 3
D)(2,6), (3,5), (4,4), (5,3), (6,2); 5

Answer 2

D

Question 3

Three representatives, if two must be male and one must be female

A)ACB, ACD, AEB, AED, CEB, CED, DEC, BEC, DEA, BEA, DCA, BCA; 12
B)ABC, CDE; 2
C)ACB, ACD, AEB, AED, CEB, CED; 6
D)ACB, ACD, AEB, AED; 4

Answer 3

C

Question 4

Solve the problem.
Construct a product table showing all possible two-digit numbers using digits from the set {1, 2, 6, 7}. List the square numbers in the table.

A){4, 9}
B){16}
C){16, 22}
D)none

Question 5

Using the 36 possibilities found in the product table for rolling two dice, list and count the outcomes for which the sum
(for both dice)is the following.
Between 7 and 10

A)(2,6), (3,6), (5,3), (4,4), (4,5); 5
B)(2,6), (6,2), (6,3), (3,6), (5,3), (3,5), (4,4), (4,5), (5,4); 9
C)(2,6), (6,2), (6,3), (3,6), (5,3), (3,5), (4,5); 8
D)(2,6), (6,2), (3,6), (6,3), (5,3), (3,5), (4,4), (4,5), (5,4), (3,4), (4,3), (6,4), (4,6), (5,5); 14

Question 6

Solve the problem.
Construct a product table showing all possible two-digit numbers using digits from the set {1, 2, 6, 7}.

Question 7

Using the 36 possibilities found in the product table for rolling two dice, list and count the outcomes for which the sum
(for both dice)is the following.
Multiple of 5

A)(1,4), (4,1), (2,3), (3,2); 4
B)(1,5), (2,5), (3,5), (4,5), (5,5), (6,5); 6
C)None
D)(1,4), (4,1), (2,3), (3,2), (4,6), (6,4), (5,5); 7

Question 8

Using the 36 possibilities found in the product table for rolling two dice, list and count the outcomes for which the sum
(for both dice)is the following.
Multiple of 3

A)(1,3), (3,1), (3,3), (2,3), (3,2), (3,4), (4,3), (5,3), (3,5), (6,3), (3,6); 11
B)(1,2), (2,1), (2,4), (4,2), (1,5), (5,1), (3,3), (3,6), (6,3), (4,5), (5,4), (6,6); 12
C)(1,2), (1,5), (3,3), (3,6), (4,5), (6,6); 6
D)(3,3), (6,6); 2

Question 9

Solve the problem.
Construct a product table showing all possible two-digit numbers using digits from the set {1, 2, 6, 7}. List the prime numbers in the table.

A){11, 17, 61, 67, 71, 77}
B){11, 17, 61, 67, 71}
C){61, 71}
D){2, 3, 7, 13}

Question 10

Using the 36 possibilities found in the product table for rolling two dice, list and count the outcomes for which the sum
(for both dice)is the following.
Greater than 10

A)(6,5), (5,6), (6,6); 3
B)(6,5), (6,6); 2
C)(6,6); 1
D)(4,6), (6,4), (5,6), (6,5), (5,5); 5

Question 11

Solve the problem.
Construct a product table showing all possible two-digit numbers using digits from the set {1, 2, 6, 7}. List the even numbers in the table.

A){12, 26, 26, 62, 66, 72, 76}
B){2, 4, 8, 12, 14}
C){62, 72}
D){12, 16, 22, 26, 62, 66, 72, 76}

Question 12

Using the 36 possibilities found in the product table for rolling two dice, list and count the outcomes for which the sum
(for both dice)is the following.
Less than 3

A)(1,1), (1,2); 2
B)(1,1), (1,2), (2, 1), (2,2); 4
C)(1,1), (1,2), (2,1); 3
D)(1,1); 1

Question 13

Three representatives, if two must be female and one must be male

A)BDA, BDC, BDE, BAD, BCD, BED;6
B)BDA, BDC, BDE, DBA, DBC; 5
C)BDA, BDC, BDE; 3
D)BDA, BDC; 2

Question 14

A president, a secretary, and a treasurer, if the president must be a woman and the other two must be men

A)BAC, BAE, DAC, DAE; 4
B)BAC, BAE, BCE, DAC, DAE, DCE, BCA, BEA, BEC, DCA, DEA, DEC; 12
C)BAC, BAE, BCE, DAC, DAE, DCE; 6
D)ABD, CBD, EBD; 3

Question 15

Use a tree diagram showing all possible results when a die is rolled twice. List the ways of getting the following result.
The first die shows a 3.

A)(3,3)
B)(3,1),(3,2),(3,4),(3,5),(3,6)
C)(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)
D)(3,1),(3,3),(3,5)

Question 16

Using the 36 possibilities found in the product table for rolling two dice, list and count the outcomes for which the sum
(for both dice)is the following.
Less than 4

A)(1,1), (1,2), (2,1); 3
B)(1,1), (1,2); 2
C)(1,1), (2,2), (1,2), (2,1); 4
D)(1,1), (1,2), (2,1), (1,3), (3,1), (2,2); 6

Question 17

Solve the problem.
Construct a product table showing all possible two-digit numbers using digits from the set {1, 2, 6, 7}. List the numbers with repeating digits.

A){11, 66, 77}
B){22, 66}
C){11, 22, 66, 77}
D)none

Question 18

Using the 36 possibilities found in the product table for rolling two dice, list and count the outcomes for which the sum
(for both dice)is the following.
Multiple of 11

A)(5,6), (6,5); 2
B)None
C)(1,1); 1
D)(5,6); 1

Question 19

A treasurer and a secretary if the two must not be the same sex

A)AB, AD, CB, CD, EB, ED, BA, DA, BC, DC, BE, DE; 12
B)AB, AC, AD, AE, BC, BD, BE, CD, CE, DE; 10
C)AB, CB, EB; 3
D)AB, AD, CB, CD, EB, ED; 6

Question 20

Four representatives

A)ABCD; 1
B)ABCD, ABCE, ACDE, ADEB, BCDE; 5
C)ABCD, ABCE, ACDE, ADEB; 4
D)ABCD, ABCE, ACDE, ADEB, BCDE, BCEA, BDEA, CABD, CEDB, DACE; 10

Question 21

Use a tree diagram showing all possible results when four fair coins are tossed. Then list the ways of getting the indicated
result.
the same on all four coins

A)hhhh, hhht, tttt, ttth
B)tttt
C)hhhh, tttt
D)hhhh

Question 22

Use a tree diagram showing all possible results when a die is rolled twice. List the ways of getting the following result.
The second die shows a 3.

A)(1,3),(2,3),(3,3),(4,3),(5,3),(6,3)
B)(1,3),(3,3),(5,3)
C)(1,3),(2,3),(4,3),(5,3),(6,3)
D)(3,3)

Question 23

Use a tree diagram showing all possible results when a die is rolled twice. List the ways of getting the following result.
The sum of the numbers showing is either 4 or 5 and one die is a 2.

A)(2,2),(2,3)
B)(2,2),(3,2)
C)(2,2),(2,3),(3,2)
D)(2,3),(3,2)

Question 24

Use a tree diagram showing all possible results when a die is rolled twice. List the ways of getting the following result.
The sum of the numbers showing is 7 and one die is a 3.

A)(3,4)
B)(4,3)
C)0
D)(3,4),(4,3)

Question 25

Determine the number of figures (of any size)in the design.
Squares (of any size)

Question 26

Use a tree diagram showing all possible results when a die is rolled twice. List the ways of getting the following result.
The first and second die are even.

A)(2,2),(4,4),(6,6)
B)(2,4),(2,6),(4,2),(4,6)(6,2),(6,4)
C)(2,2),(2,4),(2,6),(4,2),(4,4),(4,6),(6,2),(6,4),(6,6)
D)(2,2),(2,4),(2,6)

Question 27

Use a tree diagram showing all possible results when four fair coins are tossed. Then list the ways of getting the indicated
result.
exactly two tails

A)hhtt, htht, htth, thth, tthh
B)tthh
C)hhtt, htht, htth, thht, thth, tthh, ttht
D)hhtt, htht, htth, thht, thth, tthh

Question 28

Use a tree diagram showing all possible results when four fair coins are tossed. Then list the ways of getting the indicated
result.
tails on the first coin

A)thhh, thht, thtt, tthh, ttht, tttt
B)thhh, thht, thth, thtt, tthh, ttth, tttt
C)thhh, tthh, ttth, tttt
D)thhh, thht, thth, thtt, tthh, ttht, ttth, tttt

Question 29

Determine the number of figures (of any size)in the design.

Question 30

Use a tree diagram showing all possible results when a die is rolled twice. List the ways of getting the following result.
Exactly one die shows a 3.

A)(3,3)
B)(3, 1),(3, 2),(3, 3), (3, 4),(3, 5),(3, 6), (1, 3), (2, 3), (4, 3), (5, 3), (6, 3)
C)(3, 1),(3, 2),(3, 4),(3, 5),(3, 6)
D)(3, 1),(3, 2),(3, 4),(3, 5),(3, 6), (1, 3), (2, 3), (4, 3), (5, 3), (6, 3)

Question 31

Determine the number of figures (of any size)in the design.

Question 32

Use a tree diagram showing all possible results when four fair coins are tossed. Then list the ways of getting the indicated
result.
more than two tails

A)httt, thtt, ttht, ttth
B)httt, thtt, ttht, ttth, tttt
C)hhtt, htht, htth, httt, thht, thth, thtt, tthh, ttht, ttth, tttt
D)hhtt, htht, htth, thht, thth, tthh

Question 33

Use a tree diagram showing all possible results when four fair coins are tossed. Then list the ways of getting the indicated
result.
the same outcome on the first two coins

A)hhhh, hhht, hhth, hthh, hhtt, tthh, ttht, ttth, tttt
B)hhhh, hhht, hhth, hhtt, tthh, ttht, ttth, tttt
C)hhhh, hhht, hhtt, tthh, ttht, ttth, tttt
D)hhhh, hhht, hhth, hhtt

Question 34

Use a tree diagram showing all possible results when a die is rolled twice. List the ways of getting the following result.
The sum of the numbers showing is 5.

A)(2,3),(3,2)
B)(2,3),(4,1)
C)(1,4),(2,3),(3,2),(4,1)
D)(3,2),(4,1)

Question 35

Determine the number of figures (of any size)in the design.
Triangles (of any size)

Question 36

Determine the number of figures (of any size)in the design.

Question 37

Use a tree diagram showing all possible results when a die is rolled twice. List the ways of getting the following result.
The sum of the numbers showing is either 3 or 4.

A)(1,2),(2,2)
B)(1,2),(1,3),(2,2)
C)(1,2),(2,1),(1,3),(3,1),(2,2)
D)(2,1),(3,1),(2,2)

Question 38

Use a tree diagram showing all possible results when a die is rolled twice. List the ways of getting the following result.
At least one die shows a 3.

A)(3, 1),(3, 2),(3, 3), (3, 4),(3, 5),(3, 6), (1, 3), (2, 3), (4, 3), (5, 3), (6, 3)
B)(3, 1),(3, 2),(3, 4),(3, 5),(3, 6), (1, 3), (2, 3), (4, 3), (5, 3), (6, 3)
C)(3, 1),(3, 2),(3, 4),(3, 5),(3, 6)
D)(3,3)

Question 39

Use a tree diagram showing all possible results when a die is rolled twice. List the ways of getting the following result.
The sum of the numbers showing is 8.

A)(2,6),(3,5),(4,4),(5,3),(6,2)
B)(2,6),(3,5),(4,4)
C)(2,6),(3,5),(5,3),(6,2)
D)(4,4)

Question 40

Use a tree diagram showing all possible results when four fair coins are tossed. Then list the ways of getting the indicated
result.
at least two tails

A)hhtt, htht, httt, thht, thth, tthh, ttht, ttth, tttt
B)hhtt, htht, htth, httt, thht, thth, thtt, tthh, ttht, ttth, tttt
C)hhtt, htht, htth, thht, thth, tthh
D)httt, thtt, ttht, ttth, tttt

Question 41

Solve the problem.
Suppose there are 8 roads connecting town A to town B and 6 roads connecting town B to town C. In how many ways can a person travel from A to C via B?

A)14 ways
B)36 ways
C)48 ways
D)64 ways

Question 42

Evaluate the factorial expression.

Question 43

Solve the problem.
Six strangers arrive at a business seminar and each person shakes hands with every other person. How many handshakes are there?

A)18
B)20
C)15
D)30

Question 44

Evaluate the factorial expression.

Question 45

Evaluate the factorial expression.

Question 46

Evaluate the factorial expression.

Question 47

Evaluate the factorial expression.

Question 48

Solve the problem.
A computer printer allows for optional settings with a panel of four on-off switches in a row. How many different settings can be selected if no three adjacent switches can all be off?

A)13
B)12
C)14
D)10

Question 49

Solve the problem.
A sports shop sold tennis rackets in 3 different weights, 3 types of string, and 4 grip sizes. How many different rackets could be sold?

A)24 rackets
B)36 rackets
C)27 rackets
D)10 rackets

Question 50

Solve the problem.

Question 51

Evaluate the factorial expression.

Question 52

Solve the problem.
A salesperson packed 3 shirts and 5 ties. With one shirt, he could wear all 5 ties. With another shirt, he could wear 4 ties. With the third shirt, he could wear only 2 ties. How many different
Combinations did he have?

A)40 combinations
B)11 combinations
C)10 combinations
D)22 combinations

Question 53

Evaluate the factorial expression.

Question 54

Solve the problem.
A restaurant offered salads with 3 type(s)of dressing and 6 different topping(s). How many different types of salad could be ordered?

A)36
B)18
C)9
D)9

Question 55

Evaluate the factorial expression.

Question 56

Solve the problem.
In how many ways can the number 44 be written as the sum of two primes?

A)3
B)6
C)4
D)5

Question 57

A shirt company has 3 designs, each of which can be made with short or long sleeves. There are 7 color patterns available. How many different types of shirts are available from this company?

A)21 types
B)42 types
C)10 types
D)12 types

Question 58

Evaluate the factorial expression.

Question 59

Solve the problem.
How many three-digit numbers have the sum of their digits equal to 20?

A)30
B)40
C)36
D)32

Question 60

Evaluate the factorial expression.

Question 61

Solve the problem.
How many different 4-letter radio-station call letters can be made if the first letter must be K or W, repeats are allowed, but the call letters cannot end in an O?

A)35,152
B)456,976
C)33,800
D)16,900

Question 62

Solve the problem.
License plates are made using 2 letters followed by 2 digits. How many plates can be made if repetition of letters and digits is allowed?

A)456,976
B)10,000
C)67,600
D)6760

Question 63

Solve the problem.
A local department store sells carpets in 4 sizes. Each carpet comes in 3 different qualities. One of the sizes comes in 5 colors. The other sizes come in 3 colors. How many choices of carpet are there?

A)42
B)44
C)47
D)51

Question 64

Solve the problem.
How many different sequences of 4 digits are possible if the first digit must be 3, 4, or 5 and if the sequence may not end in 000? Repetition of digits is allowed.

A)2,997
B)2,000
C)1,512
D)2,999

Question 65

Solve the problem.
How many 5-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, if repetitions of digits are allowed?

A)3125
B)120
C)99,999
D)100,000

Question 66

Solve the problem.
A salesman packed 3 shirts and 6 ties. With one shirt, he could wear all 6 ties. With another shirt, he could wear 5 ties. With the third shirt, he could wear only 3 ties. How many different
Combinations did he have?

A)14
B)90
C)33
D)13

Question 67

Solve the problem.
Given a committee of 8 women and 11 men, count the number of different ways of choosing a president, a secretary, and a treasurer, if the president must be a woman and the secretary and
Treasurer must be men. Assume no one can hold more than one office.

A)440
B)5814
C)968
D)880

Question 68

Solve the problem.
Mark can remember only the first 4 digits of his friend's phone number. He also knows that the number has 7 digits and that the last digit is not a 0. If Mark were to dial all of the possible numbers
And if it takes him 21 seconds to try each one, how long would it take to try every possibility?

A)35 minutes
B)10.5 minutes
C)350.1 minutes
D)315 minutes

Question 69

Solve the problem.
A restaurant offers salads with 2 types of lettuce, 5 different toppings, and 5 different dressings. How many different salads could be ordered?

A)50
B)35
C)20
D)12

Question 70

Solve the problem.
In how many ways can 6 people line up for play tickets?

A)1
B)720
C)46,656
D)6

Question 71

Solve the problem.
How many ways can a president, vice-president, and secretary be chosen from a club with 12 members?

A)1320
B)6
C)220
D)36

Question 72

Solve the problem.
At a lumber company, shelves are sold in 4 types of wood, 2 different widths and 5 different lengths. How many different types of shelves could be ordered?

A)30
B)32
C)11
D)40

Question 73

Solve the problem.
A shoe store carries one brand of shoe in 4 different styles, 5 sizes, and 5 colors. How many different shoes are available of this one brand?

A)100
B)45
C)80
D)14

Question 74

Solve the problem.
A saleswoman packed 3 jackets and 6 skirts. With one jacket, she could wear all 6 skirts. With another jacket, she could wear 5 skirts. With the third jacket, she could wear only 2 skirts. How
Many different combinations did she have?

A)13
B)12
C)32
D)60

Question 75

Solve the problem.
A sports shop sells tennis rackets in 4 different weights, 3 types of string, and 3 grip sizes. How many different rackets could they sell?

A)10
B)21
C)36
D)48

Question 76

Solve the problem.
If 5 newborn babies are randomly selected, how many different gender sequences are possible?

A)25
B)10
C)120
D)32

Question 77

Solve the problem.

Question 78

Solve the problem.
Given a group of 8 women and 11 men, how many different ways are there of choosing one man and one woman for a committee?

A)361
B)19
C)88
D)342

Question 79

Solve the problem.
A baseball manager has 10 players of the same ability. How many 9 player starting lineups can he create?

A)10
B)362,880
C)90
D)3,628,800

Question 80

Solve the problem.
A musician plans to perform 5 selections for a concert. If he can choose from 7 different selections, how many ways can he arrange his program?

A)16,807
B)35
C)21
D)2520