Question

An investigator interviewed $100$ students to determine their preferences for the three drinks: milk (M), coffee(C) and tea (T). He reported the following: $10$ students had all the three drinks M, C, T; $20$ had M and C only; $30$ had C and T; $25$ had M and T; $12$ had M only; $5$ had C only; $8$ had T only. Then how many did not take any of the three drinks is?

• A. $20$
• B. $30$
• C. $36$
• D. $42$

Answer 1

The correct option is A $20$

Let $N_M$ be the number of students who had Milk(M) only, $N_T$ be the number of students who had Tea(T) only, $N_C$ be the number of students who had Coffee(C) only, $N_{MC}$ is the number of students who had Milk(M)&Coffee(C) but no Tea(T), $N_{MT}$ is the number of students who had Milk(M)&Tea(T) but no Coffee(C), $N_{TC}$ is the number of students who had Tea(T)&Coffee(C) but no Milk(M) and $N_{MCT}$ is the number of students who had all the three drinks Milk(M), Coffee(C), Tea(T).

To find the number of students who did not take any of the drink we have to take away students who take any of the drink from $100$ students.

Students who take any of the drink are as follows:

$N_M=12$, $N_C=5$, $N_T=8$, $N_{MCT}=10$.

$N_{MC}=20-N_{MCT}=20-10=10$.

$N_{MT}=25-N_{MCT}=25-10=15$.

$N_{TC}=30-N_{MCT}=30-10=20$.

Now, number of students who take any of the drink will be:

$N_M+N_C+N_T+N_{MC}+N_{MT}+N_{TC}+N_{MCT}=12+5+8+10+15+20+10=80$.

Finally, the number of students who did not take any of the drink is $100-80=20$.

Hence, $20$ students did not take any of the three drinks.