Question 1

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

Accepted Answer

The last four digits can vary, and since each digit can be any number from 0 to 9, there are 10^4 = 10,000 possible combinations for the last four digits.

Question 2

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

Accepted Answer

The answer of Find how many outcomes are possible for...

Question 3

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

Accepted Answer

The answer of Professor Oger must choose between an extra...

Question 4

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

Accepted Answer

The answer of An experiment requires a sequence of three...

Question 5

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

Accepted Answer

The answer of An experiment requires a sequence of three...

Question 6

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

Accepted Answer

The answer of A test requires that you answer first...

Question 7

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

Accepted Answer

The answer of How many three-letter sequences are possible that...

Question 8

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

Accepted Answer

The answer of A test requires that you answer either...

Question 9

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

Accepted Answer

To alternate sides, the sequence must start with an odd-numbered cylinder (right side) and then an even-numbered cylinder (left side), continuing this pattern. There are 4 odd-numbered cylinders (1, 3, 5, 7) and 4 even-numbered cylinders (2, 4, 6, 8). The number of ways to arrange the odd cylinders is 4! (24 ways), and the number of ways to arrange the even cylinders is also 4! (24 ways). Therefore, the total number of good firing sequences is 4! * 4! = 576. However, since the sequence can start with either an odd or even cylinder, we multiply by 2, giving 2 * 576 = 1,152 sequences.

Question 10

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

Accepted Answer

To represent 4,096 different characters, we need to use 12 bits because:
2^12 = 4,096

Question 11

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

Accepted Answer

The lowest number of calendars is achieved by finding the minimum number of different patterns that can represent every possible combination of days for each month. This can be done using what is known as the "Doomsday algorithm". The algorithm relies on 14 different calendars, which correspond to the 14 different possible ways that the Doomsday falls on the days of the week for a given year. Therefore, the lowest number of calendars is 14, which is option D.

Question 12

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

Accepted Answer

For Alternative 1, the total outcomes are 6 (Step 1) * 2 (Step 2) = 12. For Alternative 2, the total outcomes are 3 (Step 1) * 2 (Step 2) * 2 (Step 3) = 12. Adding both alternatives gives 12 + 12 = 24 outcomes.

Question 13

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

Accepted Answer

The answer of A surgical procedure requires choosing among three...

Question 14

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

Accepted Answer

There are only two days on which February 11 falls on a Monday or Tuesday in any given year. Those days are Monday and Tuesday. So the possibilities are: 
1) February 11 is on Monday and there are 28 days in February. 
2) February 11 is on Tuesday and it is a leap year, so there are 29 days in February. 
There are no other possibilities in any given year. However, we are not given a year, so we can't definitively say that there are only 2 calendars possible. But the answer choices suggest that we select the most likely option, which is 4 calendars.

Question 15

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

Accepted Answer

The answer of A telephone number consists of a sequence...

Question 16

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

Accepted Answer

There are 2 outcomes in Step 1 and 3 outcomes in Step 2, so there are 2 * 3 = 6 possible outcomes overall.

Question 17

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

Accepted Answer

The answer of A surgical procedure requires choosing among three...

Question 18

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

Accepted Answer

The answer of Dripping wet after your shower, you have...

Question 19

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

Accepted Answer

For gummy candy, there are 3 sizes and 5 types, so the choices are 3 sizes * 5 types = 15. For licorice nibs, there are 3 sizes and 2 types, so the choices are 3 sizes * 2 types = 6. Adding them together, 15 (gummy) + 6 (licorice) = 21 choices.

Question 20

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

Accepted Answer

To find the total number of outcomes, we need to multiply the number of outcomes at each step. So, the total number of outcomes would be:

3 (outcomes in step 1) * 5 (outcomes in step 2) * 4 (outcomes in step 3) * 4 (outcomes in step 4) = 240

Therefore, the answer is B) 240.

Quiz 7: Sets and Counting

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

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

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 ?

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

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 ?

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 ?

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 ?

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

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

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

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 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 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 ?

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