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

Solution:

Let D denotes dancing, P denotes painting and S denotes singing.

$\begin{matrix} ∴ & n (D \cup P \cup S) = 265, \\ n (S) = 200, n (D) = 110, n (P) = 55, \\ and & n (S \cap D) = 60, n (S \cap P) = 30 \\ ∵ & n (D \cup P \cup S) = n (D) + n (P) + n (S) - n (D \cap P) - n (P \cap S) - n (S \cap D) + n (D \cap P \cap S) \\ ∴ & 265 = 110 + 55 + 200 - n (D \cap P) - 30 - 60 + 10 \\ \Rightarrow & 265 = 285 - n (D \cap P) \Rightarrow n (D \cap P) = 20 \end{matrix}$

$∴$ Persons who like only dancing and painting $= n (D \cap P) - n (D \cap P \cap S) = 20 - 10 = 10$

Solution:

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