Question

Solve the following problem and verify the data in this case by drawing Venn diagram. In a group of 50 persons, 30 like tea, 25 like coffee and 16 like both. How many like neither tea nor coffee?

Answer 1

Let the total number of persons be $n\left(U\right)$, number of person who like tea be $n\left(T\right)$ and number of person who like coffee be $n\left(C\right)$.

Then, the number of person who like both tea and coffee is $n(T\cap C)$ and the number of person who like either tea or coffee is $n(T\cup C)$.

We know that $n(A\cup B)=n\left(A\right)+n\left(B\right)-n(A\cap B)$ where $A$ and $B$ are the respective sets.

Here, it is given that total number of persons are $50$ of which $30$ like tea, $25$ like coffee and $16$ like both, that is $n\left(U\right)=50,n\left(T\right)=30,n\left(C\right)=25$ and $n(T\cap C)=16$ and therefore,

$n(T\cup C)=n\left(T\right)+n\left(C\right)-n(T\cap C)=30+25-16=55-16=39$

Thus, $39$ persons like either tea or coffee.

Now, the number of persons who like neither tea nor coffee is $n\left(U\right)-n(T\cup C)$ that is $50-39=11$

Hence, $11$ persons like neither tea nor coffee.