Question

In a survey, it was found that 21 people like product $A$, 26 like product $B$ and 29 like product $C$. If 14 people like products $A$ and $B,12$ people like products $C$ and $A,14$ people like products $B$ and $C$ and 8 like all the three products, then number of people who liked product $C$ only, is

• A. 7
• B. 15
• C. 11
• D. 9

Answer 1

Answer: (C)

Let $A$ be the set who like product $A,B$ be the set who like product $B$ and $C$ be the set who like product $C$.

Then,
$n(A)=21,n(B)=26,n(C)=29$
$n(A\cap B)=14,n(B\cap C)=14$
$n(A\cap C)=12,n(A\cap B\cap C)=8$
$a+b+e+f=\mathrm{21....}$(i)
$b+c+d+e=\mathrm{26....}$(ii)
$d+e+f+g=\mathrm{29.....}$(iii)
$b+e=\mathrm{14.....}$(iv)
$e+d=\mathrm{14....}$(v)
$f+e=\mathrm{12....}$(vi)
$e=\mathrm{8....}$(vii)
Substituting the values of $e$ in Eqs. (iv), (v) and (vi), we get
$b=6,d=6$ and $f=4$
Substituting the values of $d,e$ and $f$ in Eq. (iii), we get
$\therefore 6+8+4+g=29$
$\therefore g=11$
Number of people who like product $C$ only, is $11$.
Alternate Method
$n\left(C\cap A^{\prime }\cap B^{\prime }\right)=n(C)-n(C\cap A)-n(C\cap B)$
$+n(A\cap B\cap C)$
$=29-12-14+8$
$=37-26$
$=11$