Question

One hundred people are surveyed and asked if they trust candidate A or candidate B or candidate C in a local election. People are allowed to indicate more than one candidate and they are allowed to indicate no candidates. The results include the following:

34 people trust candidate A
44 people trust candidate B
48 people trust candidate C
7 people trust candidates both A and B
12 people trust candidates both A and C
17 people trust candidates both B and C
5 people trust candidates all three A and B and C

How many of the people surveyed trust NONE of the candidates?

Answer 1

5 people trust none of the candidates

Explanation:

Given:

n = 100 people

34 people trust candidate A

44 people trust candidate B

48 people trust candidate C

7 people trust candidates both A and B

12 people trust candidates both A and C

17 people trust candidates both B and C

5 people trust candidates all three A and B and C

Number of people that trust only A and B = 7 - 5 = 2

Number of people that trust only A and C = 12 - 5 = 7

Number of people that trust only B and C = 17 - 5 = 12

Number of people that trust only A = 34 - 7 - 5 - 2 = 20

Number of people that trust only B = 44 - 12 - 5 - 2 = 25

Number of people that trust only C = 48 - 7 - 5 - 2 = 24

Number of people that trust at least one candidate

= 20 + 24 + 25 + 7 + 2 + 12 + 5

= 95

The number of people that trust none of the candidates will be:

100 - (20+24+25+7+2+12+5)

= 100 - 95

= 5

5 people trust none of the candidates.

Answer 2

5 people trust none of the candidates

Explanation:

To know how many people surveyed trust none of the candidates we need to find:

• People that trust all three candidates: 5
• People that just trust candidate B and C: This is equal to people that trust candidate B and C less people that trust all three candidates. So it is equal to: 17 - 5 = 12
• People that just trust candidate A and C: This is equal to people that trust candidate A and C less people that trust all three candidates. So it is equal to: 12 - 5 = 7
• People that just trust candidate A and B: This is equal to people that trust candidate A and B less people that trust all three candidates. So it is equal to: 7 - 5 = 2
• People that just trus candidate C: This is equal to the people that trust candidate C less people that just trust candidate B and C less people that just trust candidate A and C less people that trust all three candidates. So, it is equal to: 48 - 12 - 7 - 5 = 24
• People that just trus candidate B: This is equal to the people that trust candidate B less people that just trust candidate B and C less people that just trust candidate A and B less people that trust all three candidates. So, it is equal to: 44 - 12 - 2 - 5 = 25
• People that just trus candidate A: This is equal to the people that trust candidate A less people that just trust candidate A and C less people that just trust candidate A and B less people that trust all three candidates. So, it is equal to: 34 - 7 - 2 - 5 = 20

Therefore, we can calculate how many people surveyed trust at least one candidate by the sum of the previous quantities as:

5 + 12 + 7 + 2 + 24 + 25 + 20 = 95

Finally, there are 100 people surveyed and 95 people trust at least one candidate, so 5 people trust none of the candidates.