Question

An investigator interviewed 100 students to determine their preferences for the three drinks milk,coffee and tea.He reported the following ,10 students had all the three drinks,20 had milk and coffee,30 had coffee and tea,25 had milk and tea,12 had milk only.5 had coffee only,8 had tea only.The number of students that did not take any of the three drinks is ?

Answer 1

Hint: Represent the given three drinks had by the students by any known notation.Number of students who did not take any of the drinks is equal to the total number of students minus students who take any of the drink.And find the required solution according to the statement.

Complete step by step answer:
Let us denote that,
The number of students who had Milk only = \[{{N}_{M}}\]
The number of students who had Tea only = \[{{N}_{T}}\]
The number of students who had Coffee only = \[{{N}_{C}}\]
The number of students who had Milk and Coffee but not Tea = \[{{N}_{MC}}\]
The number of students who had Milk and Tea but not Coffee = \[{{N}_{MT}}\]
The number of students who had Tea and Coffee but not Milk = \[{{N}_{TC}}\]
The number of students who had all the three drinks Milk,Coffee,Tea= \[{{N}_{MCT}}\]
Now to find the number of students who did not take any of the drinks we have to eliminate the students who take any of the drinks from 100 students.
Now the students who take any of the drink is as follows,
\[{{N}_{M}}=12;{{N}_{T}}=8;{{N}_{C}}=5;{{N}_{MCT}}=10;\]
Now,
\[\begin{align}
  & {{N}_{MC}}=20-{{N}_{MCT}}=20-10=10 \\
 & {{N}_{MT}}=25-{{N}_{MCT}}=25-10=15 \\
 & {{N}_{TC}}=30-{{N}_{MCT}}=30-10=20 \\
\end{align}\]
  Number of students who take any of the drink
 ⟹\[{{N}_{M}}+{{N}_{C}}+{{N}_{T}}+{{N}_{MC}}+{{N}_{MT}}+{{N}_{TC}}+{{N}_{MCT}}\]
⟹ \[12+8+5+10+15+20+10\]
⟹ \[80\]
Therefore,number of students who did not take any of the drink \[=100-80=20\]

Note:
The above given problem can also be solved by using venn diagram method or by using
probability formulae. Here the students make a mistake in calculating the students who
take any both drinks but not one drink is equal to the students having both drinks minus
student having all the three drinks.