Question

In a shop, $380$ people buy socks, $150$ people buy shoes, and $200$ people buy belts. If there are a total of $580$ people who bought either socks or shoes or belts and only $30$ people bought all the three things? So, how many people bought exactly two things?

• A. $90$
• B. $135$
• C. $45$
• D. $150$

Answer 1

The correct option is A $90$

Let $S,H,$ and $B$ represent the set of numbers of people who bought socks, shoes, and belts, respectively.
So, $n\left(S\right)=380,n\left(H\right)=150,n\left(B\right)=200$
$n(S\cup H\cup B)=580,n(S\cap H\cap B)=30$
So, $n(S\cup H\cup B)=n\left(S\right)+n\left(H\right)+n\left(B\right)–n(S\cap H)–n(H\cap B)–n(B\cap S)+n(S\cap H\cap B)$,
Now, we put values given in the formula,
$580=380+150+200-n(S\cap H)–n(H\cap B)–n(B\cap S)+30$
This gives that,
$n(S\cap H)+n(H\cap B)+n(B\cap S)=180$
But this includes the number of people who bought all the three items also. So, we have to deduct this number of people from it. Let, $n(S\cap H\cap B)=a$
As we can see from the Venn diagram,
$n(S\cap H)–a+n(H\cap B)–a+n(B\cap S)–a=$ the required number $n(S\cap H)+n(H\cap B)+n(B\cap S)-3a$
$180–90=90$
Hence, $90$ people bought exactly two things.