Question

There is a group of 265 persons who like either singing or dancing or painting. In this group, 200 like singing, 110 like dancing and 55 like painting. If 60 persons like both singing and dancing, 30 like both singing and painting and 10 like all three activities, then the number of persons who like only dancing and painting is
A.10
B.20
C.30
D.40

Answer 1

Hint: Here in this question, we have to find how many people only like dancing and painting in a group of 265 persons. To find this we can use the union and intersection concept of set theory and also using basic arithmetic operations like addition and subtraction on simplification we get the required solution.

Complete step-by-step answer:
Consider the given question:
Let the number of people who like singing be $$n\left( S \right) = 200$$.
The number of people who like dancing is $$n\left( D \right) = 110$$.
The number of people who like painting is $$n\left( P \right) = 55$$.
Given the total number of groups of persons be $$n\left( {S \cup D \cup P} \right) = 265$$.
The number of people who like both singing and dancing $$ = n\left( {S \cap D} \right) = 60$$
The number of people who like both singing and painting $$ = n\left( {S \cap P} \right) = 30$$ and
The number of people who like all the three activities like singing, dancing and painting $$ = n\left( {S \cap D \cap P} \right) = 10$$
We have to find the number of people who only like dancing and painting?
We know that, for any two sets A and B, then$$n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)$$
 For the set S, D and P, then
$$n\left( {S \cup D \cup P} \right) = n\left( S \right) + n\left( D \right) + n\left( P \right) - n\left( {S \cap D} \right) - n\left( {D \cap P} \right) - n\left( {P \cap S} \right) + n\left( {S \cap D \cap P} \right)$$
On substituting, we have
$$ \Rightarrow \,\,\,265 = 200 + 110 + 55 - 60 - n\left( {D \cap P} \right) - 30 + 10$$
$$ \Rightarrow \,\,\,265 = 285 - n\left( {D \cap P} \right)$$
On rearranging, we have
$$ \Rightarrow \,\,\,n\left( {D \cap P} \right) = 285 - 265$$
$$ \Rightarrow \,\,\,n\left( {D \cap P} \right) = 20$$
Now, the number of persons who like only dancing and painting $$ = \,n\left( {D \cap P} \right) - n\left( {S \cap D \cap P} \right)$$
$$ \Rightarrow \,\,\,20 - 10$$
$$\therefore \,\,\,10$$
Hence, the number of people who only like dancing and painting is 10.
Therefore, option (A) is the correct answer.
So, the correct answer is “Option A”.

Note: The question belongs to the concept of set. the union (denoted by$$ \cup $$) of a collection of sets is the set of all or total elements in the collection and the intersection (denoted by $$ \cap $$) of a collection of sets If two sets A and B then $$A \cap B$$is the set containing all elements of A that also belong to B. While simplifying we use the simple arithmetic operations and hence we obtain the result.