vousetmoiec

2021-12-04

A survey of 70 college students showed the following data: 42 had a car; 50 had a TV; 30 had a bicycle; 17 had a car and a bicycle; 35 had a car and a TV; 25 had a TV and a bicycle; 15 had all three. How many students had none of the three items?

Alfonso Miller

Beginner2021-12-05Added 20 answers

Step 1

According to a survey of 70 college students, 42 had a car, 50 had a TV, 30 had a bicycle, 17 owned both a car and a bicycle, 35 owned both a car and a television,

Step 2

That is, $n\left(C\right)=43,n\left(T\right)=50,n\left(B\right)=30,n\left(C\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}B\right)=17,n\left(C\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}T\right)=35,n\left(T\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}B\right)=25\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}n\left(C\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}T\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}B\right)=15$.

Total students is $n=70$.

Thus, the number of students, who had either car or TV or bicycle is,

$n\left(C\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{or}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}T\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{or}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}B\right)=n\left(C\right)+n\left(T\right)+n\left(B\right)-n\left(C\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}B\right)-n\left(C\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}T\right)-n\left(T\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}B\right)+n\left(C\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}T\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}B\right)=43+50+30-17-35-25+15=61$.

Thus, the number of students, had none of the three items is $n-n\left(C\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{or}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}R\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{or}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}B\right)=70-61=9$.