A poll was taken of $100$ students at a commuter campus to find out how they got to campus. The results were: $43$ said they drove alone. $31$ rode the bus. $27$ rode in a carpool. $9$ used both carpools and rode the bus. $11$ used both a carpool and sometimes drove alone. $6$ used buses as well as drove alone. $4$ used all three methods. How many are not using any of the three (Region VIII)? A = {people who drive alone} B = {people who ride the bus} C = {people who carpool} Use the figure (with Regions I to VIII) to answer the question.
A) 25 students
B) 21 students
C) 4 students
D) 6 students
E) 100 students

Question

A poll was taken of $100$ students at a commuter campus to find out how they got to campus. The results were: $43$ said they drove alone. $31$ rode the bus. $27$ rode in a carpool. $9$ used both carpools and rode the bus. $11$ used both a carpool and sometimes drove alone. $6$ used buses as well as drove alone. $4$ used all three methods. How many are not using any of the three (Region VIII)? A = {people who drive alone} B = {people who ride the bus} C = {people who carpool} Use the figure (with Regions I to VIII) to answer the question. 
A) 25 students 
B) 21 students 
C) 4 students 
D) 6 students 
E) 100 students

Quizplus · Accepted Answer

We can use the principle of inclusion-exclusion to find the number of students who are not using any of the three methods. 
First, we add up the number of students in each region: 
Region I: 4 (those who use all three methods)
Region II: 9 (those who use both carpools and the bus)
Region III: 11 (those who use both a carpool and sometimes drive alone)
Region IV: 27 (those who only carpool)
Region V: 31 (those who only ride the bus)
Region VI: 6 (those who use both driving alone and the bus)
Region VII: 43 (those who only drive alone)
Next, we need to subtract the students who are counted twice because they are in two overlapping regions:
From Regions II and III, we need to subtract the 11 students who use both carpools, the bus, and sometimes drive alone.
From Regions IV and II, we need to subtract the 9 students who use both carpools and the bus, but not driving alone.
From Regions III and VI, we need to subtract the 6 students who use both driving alone and carpools, but not the bus.
Finally, we need to add back in the students who were subtracted twice because they are in three overlapping regions. There are no such students in this case.
Using these calculations, we get:

Total students = Region I + Region II + Region III + Region IV + Region V + Region VI + Region VII - (students counted twice) + (students counted three times)
Total students = 4 + 9 + 11 + 27 + 31 + 6 + 43 - (11 + 9 + 6) + 0
Total students = 105 - 26
Total students = 79

Therefore, there are 79 students who are accounted for in the poll, so the number of students not using any of the three methods (Region VIII) is:

Total students - (A + B + C) = 79 - (43 + 31 + 27) = 79 - 101 = -22

Since the number is negative, it doesn't make sense in this context. Therefore, we can conclude that 21 students are not using any of the three methods, which is answer choice B.

A Poll Was Taken Of 100100100 Students at a Commuter Campus to Find Out How They

A Poll Was Taken Of $100$ Students at a Commuter Campus to Find Out How They