Deck 57: Counting Principles

Question 1

Determine the number of ways a computer can randomly generate the odd integers from 1 through 26. ​

A)11
B)15
C)17
D)13
E)9

Answer 1

13

Question 2

In how many ways can a 22-question true-false exam be answered? (Assume that no questions are omitted. ) ​

A)4,194,299
B)484
C)4,194,294
D)4,194,314
E)​4,194,304

Answer 2

​4,194,304

Question 3

Evaluate ${ } _ { n } P _ { r }$ .? $${ } _ { 2 } P _ { 2 }$$ ?

A)2
B)20
C)30
D)6
E)12

Answer 3

2

Question 4

Determine the number of ways a computer can randomly generate two distinct integers whose sum is 13 from 1 through 17 in increasing order. ​

A)6
B)2
C)10
D)8
E)4

Question 5

Determine the number of ways a computer can randomly generate the integers that is divisible by 4 from 1 through 40.

A)14
B)12
C)10
D)6
E)8

Question 6

A college student is preparing a course schedule for the next semester.The student may select one of five mathematics courses,one of two science courses,and one of five courses from the social sciences and humanities.How many schedules are possible? ​

A)55
B)45
C)60
D)50
E)40

Question 7

How many three-digit numbers can be formed under the following condition? ​
The number is at least 400.
​

A)590
B)610
C)620
D)580
E)600

Question 8

Determine the number of ways a computer can randomly generate the even integers from 1 through 30. ​

A)17
B)13
C)15
D)11
E)19

Question 9

Ten people are boarding an aircraft.Two have tickets for first class and board before those in the economy class.In how many ways can the ten people board the aircraft? ​

A)240
B)48
C)80,640
D)12
E)1,440

Question 10

How many three-digit numbers can be formed under the following condition? ​
The leading digit cannot be zero.
​

A)800
B)​500
C)​900
D)​600
E)​700

Question 11

How many three-digit numbers can be formed under the following condition? ​
The leading digit cannot be zero and the number must be even.
​

A)900
B)450
C)350
D)400
E)550

Question 12

Determine the number of ways a computer can randomly generate the prime integers from 1 through 40. A prime integer

A)10
B)14
C)12
D)8
E)16

Question 13

In how many ways can a seven-question true-false exam be answered? (Assume that no questions are omitted. ) ​

A)138
B)128
C)133
D)118
E)123

Question 14

Determine the number of ways a computer can randomly generate the integer that is divisible by 3 from 1 through 42.

A)18
B)12
C)10
D)16
E)14

Question 15

Determine the number of ways a computer can randomly generate two distinct integers whose sum is 20 from 1 through 25 in increasing order. ​

A)7
B)5
C)11
D)13
E)9

Question 16

A college needs two additional faculty members: a chemist and a statistician.In how many ways can these positions be filled if there are three applicants for the chemistry position and four applicants for the statistics position?

A)10
B)11
C)16
D)14
E)12

Question 17

Evaluate ${ } _ { n } P _ { r }$ .? $${ } _ { 7 } P _ { 3 }$$ ?

A)210
B)336
C)720
D)35
E)504

Question 18

How many four-digit numbers can be formed under the following condition? ​
The leading digit cannot be zero and the number must be less than 7,000.
​

A)5,800
B)6,000
C)5,700
D)5,600
E)5,900

Question 19

A customer can choose one of two amplifiers,one of three compact disc players,and one of five speaker models for an entertainment system.Determine the number of possible system configurations. ​

A)40
B)35
C)​30
D)25
E)20

Question 20

Determine the number of ways a computer can randomly generate the integers that is greater than 8 from 1 through 12.

A)8
B)6
C)0
D)4
E)2

Question 21

Evaluate ${ } _ { n } P _ { r }$ .? ${}_{100} P _ { 3 }$ ?

A)94,109,400
B)161,700
C)970,200
D)858,277,728,000
E)9,034,502,400

Question 22

There are six processes involved in assembling a product,and these processes can be performed in any order.The management wants to test each order to determine which is the least time-consuming.How many different orders will have to be tested? ​

A)1,220
B)720
C)770
D)820
E)1,720

Question 23

Evaluate ${ } _ { n } P _ { r }$ .? ${}_{100} P _ { 5 }$ ?

A)94,109,400
B)970,200
C)75,287,520
D)9,034,502,400
E)858,277,728,000

Question 24

At a high school cafeteria,diners can choose one vegetable from a choice of 3 vegetables,one meat from a choice of 3 meats,one serving of bread from among 2 breads,and a dessert from among 2 desserts.How many meal configurations are possible?

A)18
B)4
C)36
D)10
E)12

Question 25

A baseball coach is creating a nine-player batting order by selecting from a team of 19 players.How many different batting orders are possible? ​

A)33,522,128,640
B)33,523,128,640
C)33,522,628,640
D)33,572,128,640
E)33,532,128,640

Question 26

Eight sprinters have qualified for the finals in the 100-meter dash at the NCAA national track meet.In how many ways can the sprinters come in first,second,and third? (Assume there are no ties. ) ​

A)486
B)386
C)436
D)336
E)536

Question 27

Evaluate ${ } _ { n } P _ { r }$ .? ${ } _ { 7 } P _ { 4 }$ ?

A)5,040
B)1,680
C)3,024
D)35
E)840

Question 28

Evaluate ${ } _ { n } C _ { r }$ using the formula from this section.? ${ } _ { 5 } C _ { 3 }$ ?

A)84
B)60
C)10
D)20
E)35

Question 29

Evaluate ${ } _ { n } C _ { r }$ using the formula from this section.? ${ } _ { 5 } C _ { 2 }$ ?

A)20
B)28
C)10
D)21
E)15

Question 30

Evaluate ${ } _ { n } P _ { r }$ .? ${ } _ { 16 } P _ { 8 }$ ?

A)57,657,600
B)524,160
C)5,765,760
D)518,918,400
E)12,870

Question 31

Evaluate ${ } _ { n } P _ { r }$ .? ${}_{40} P _ { 5 }$ ?

A)2,193,360
B)658,008
C)59,280
D)78,960,960
E)2,763,633,600

Question 32

Evaluate ${ } _ { n } C _ { r }$ using the formula from this section.? ${ } _ { 11 } C _ { 0 }$ ?

A)11
B)1
C)55
D)165
E)330

Question 33

Determine the number of ways a computer can randomly generate an integer between 10 and 20 (inclusive)that is a composite number (a composite number is a number that is not a prime number).

A)​8
B)​6
C)7
D)6
E)9

Question 34

Evaluate ${ } _ { n } C _ { r }$ using the formula from this section.? ${ } _ { 6 } C _ { 1 }$ ?

A)8
B)7
C)9
D)10
E)6

Question 35

From a pool of 12 candidates,the offices of president,vice-president,secretary,and treasurer will be filled.In how many different ways can the offices be filled? ​

A)12,380
B)13,380
C)12,880
D)13,880
E)11,880

Question 36

Evaluate ${ } _ { n } C _ { r }$ using the formula from this section.? ${ } _ { 48 } C _ { 5 }$ ?

A)1,712,304
B)377,348,994
C)12,271,512
D)73,629,072
E)205,476,480

Question 37

Evaluate ${ } _ { n } C _ { r }$ using the formula from this section.? ${ } _ { 25 } C _ { 4 }$ ?

A)177,100
B)303,600
C)480,700
D)12,650
E)53,130

Question 38

In how many ways can seven children posing for a photograph line up in a row? ​

A)5,040
B)5,140
C)5,090
D)6,040
E)5,540

Question 39

Evaluate ${ } _ { n } P _ { r }$ .? ${ } _ { 19 } P _ { 2 }$ ?

A)420
B)380
C)342
D)462
E)171

Question 40

In how many ways can seven people sit in a seven-passenger car? ​

A)6,040
B)5,090
C)5,040
D)5,140
E)5,540

Question 41

Find the number of diagonals of a dodecagon (12 sides).(A line segment connecting any two nonadjacent vertices is called a diagonal of the polygon. )

A)24
B)12
C)144
D)132
E)54

Question 42

?Evaluate the expression.? $C ( 5,2 )$ ?

A)?60
B)?120
C)?2
D)?20
E)?10

Question 43

?Evaluate the expression.? $P ( 8,5 )$ ?

A)?48
B)?6
C)?6,720
D)?80,640
E)?none of the above

Question 44

A combination lock will open when the right choice of three numbers (from 1 to 35)is selected.How many different lock combinations are possible?

A)1,225
B)42,875
C)128,625
D)35
E)105

Question 45

Evaluate the expression.​​

Question 46

Which scenario(s)below should be counted using permutations? I.the number of ways the gold,silver,and bronze medal winners can be chosen in an Olympic track-and-field event (assume that there are no ties)
II.the number of ways 3 people can be chosen to sit from among 8 people
III.the number of ways to choose 5 lottery numbers
IV.the number of ways to roll a total of 7 when rolling a pair of dice
V.the permutation may not be used in any scenarios

A)​II only
B)​I only
C)​V only
D)​IV only
E)​III only

Question 47

Evaluate: ${ } _ { 15 } P _ { 5 }$

A)32,760
B)75
C)undefined
D)360,360
E)3,003

Question 48

Evaluate the expression.? $\left( \begin{array} { l } 96 \\95\end{array} \right)$ ?

A)86
B)1
C)96
D)95
E)-9

Question 49

Twelve weightlifters are competing in the dead-lift competition.In how many ways can the weightlifters finish first,second,and third (no ties)?

A)1,728
B)6
C)36
D)3
E)1,320

Question 50

​Evaluate the expression.​

Question 51

There are 15 patients in Dr.Ziglar's waiting room.Dr.Ziglar can see 6 patients before lunch.In how many different orders can Dr.Ziglar see 6 of the patients before lunch?

A)90
B)6
C)1,816,214,400
D)3,603,600
E)5,005

Question 52

Solve for n.? $56 ^ { . }~ {}_{ n - 1} P _ { 6 } = { } _ { n + 1 } P _ { 7 }$

A)no solution
B)?n = 7
C)?n = 7,48
D)?n = 8,47
E)?n = 48

Question 53

How many 3-digit numbers can be formed if the leading digit cannot be zero?

A)​900
B)100
C)9999
D)901
E)999

Question 54

Evaluate using a graphing utility: ${ } _ { 15 } P _ { 5 }$

A)3,003
B)32,760
C)360,360
D)75
E)undefined

Question 55

Find the number of distinguishable permutations of the group of letters. G,A,U,S,S

A)​15
B)​5
C)​120
D)​60
E)​30