Question

In a certain group of 36 people, 18 are wearing hats and 24 are wearing sweaters. If six people are wearing neither a hat nor a sweater, then how many people are wearing both a hat and a sweater?
\[\begin{align}
  & A.30 \\
 & B.22 \\
 & C.12 \\
 & D.8 \\
\end{align}\]

Answer 1

Hint: In this question, we are given the number of total people, number of people who are wearing hats, number of people who are wearing sweaters and number of people who are neither wearing hats nor sweaters. We need to find a number of people who are wearing both hats and sweaters. For this, we will use property of number of elements in set given by $n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right)$ where n(A) represent number of elements in A, n(B) represent number of elements in B, $n\left( A\cup B \right)$ represent number of elements in A or B, $n\left( A\cap B \right)$ represent number of elements in both A and B.

Complete step-by-step answer:
Here we are given a number of total people as 36.
Number of people who are wearing hats are equal to 18.
Number of people who are wearing sweaters are equal to 24.
Number of people who are neither wearing hats nor sweaters are equal to 6.
We need to find the number of people who are wearing both hats and sweaters.
Let us suppose H as the set of people wearing hats and S as the set of people wearing sweaters.
According to the question, we have n(H) = 18 and n(S) = 24.
And we need to find $n\left( H\cap S \right)$ which represents elements of H and S both (people wearing both hats and sweaters).
As we know, for two sets A and B we have $n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right)$.
So we need to find $n\left( A\cup B \right)$ first.
We are given total people as 36 and out of them 6 people are neither wearing hats nor sweaters. So we can say the remaining people could be wearing either a hat or sweater or both. So we get 36-6 = 30 people who are wearing a hat or sweater or both a hat and sweater.
So we conclude that $n\left( H\cup S \right)=30$ which represents elements of H or S. (People wearing hat or sweater or both).
Putting the values in the formula given by $n\left( H\cup S \right)=n\left( H \right)+n\left( S \right)-n\left( H\cap S \right)$ we get:
\[\begin{align}
  & \Rightarrow 30=24+18-n\left( H\cap S \right) \\
 & \Rightarrow n\left( H\cap S \right)=24+18-30 \\
 & \Rightarrow n\left( H\cap S \right)=42-30 \\
 & \Rightarrow n\left( H\cap S \right)=12 \\
\end{align}\]
Hence 12 people are wearing both hats and sweaters.

So, the correct answer is “Option C”.

Note: Students can make mistakes of taking 36 as $n\left( H\cup S \right)$ but they should know that 6 people are not included in set H or S so we have not counted them. Students can get confused between $\cap \text{ and }\cup $. They can memorize it by learning that intersection sign $\cap $ is used for 'and' and union sign $\cup $ is used for 'or'.