Question

In a survey it was found that $21$ people liked product $A,26$ liked product $B$ and $29$ liked product $C$. If $14$ people liked products $A$ and $B$, $12$ people liked products $C$ and $A$, $14$ people liked products $B$ and $C$ and $8$ liked all the three products. Find how many liked product $C$ only.

Answer 1

Given data
Number of people who liked product $A=n\left(A\right)=21$
Number of people who liked product $B=n\left(B\right)=26$
Number of people who liked product $C=n\left(C\right)=29$
Number of people who liked product $A$ and $B=n(A\cap B)=14$
Number of people who liked product $C$ and $A=n(C\cap A)=12$
Number of people who liked product $B$ and $C=n(B\cap C)=14$
Number of people who liked all three products $A,B$ and $C=n(A\cap B\cap C)=8$

Draw a Venn diagram
Let $a$ denote the number of people who liked product $A$ and $B$ but not $C$.
Let $b$ denote the number of people who liked product $A$ and $C$ but not $B$.
Let $c$ denote the number of people who liked product $B$ and $C$ but not $A$.
Let $d$ denote the number of peoplewho liked all three products $A,B$ and $C$

$d=n(A\cap B\cap C)=8$
$n(A\cap C)=12$
$\Rightarrow b+d=12$
Putting $d=8$
$b+8=12$
$\Rightarrow b=4$

Similarly, $n(B\cap C)=14$
$\Rightarrow c+d=14$
$\Rightarrow c=6$

So, $b=4,c=6$ and $d=8$

Step 4:
Solve for people who liked product $C$ only.
Number of people who liked product $C$ only $=n\left(C\right)-b-d-c$
$=n\left(C\right)-(b+d+c)$
$=29-(4+8+6)$
$=11$

Final Answer:
Hence, number of people who like product $C$ only is $11$.