There is a group of 265 persons who like eith | vedclass

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(A) Let $S$, $D$, and $P$ denote the sets of persons who like singing, dancing, and painting, respectively. Given: $n(S \cup D \cup P) = 265$, $n(S) =

Vedclass · Accepted Answer

$10$

Vedclass · Answer

(A) Let $S$, $D$, and $P$ denote the sets of persons who like singing, dancing, and painting, respectively. Given: $n(S \cup D \cup P) = 265$, $n(S) = 200$, $n(D) = 110$, $n(P) = 55$, $n(S \cap D) = 60$, $n(S \cap P) = 30$, and $n(S \cap D \cap P) = 10$. Using the principle of inclusion-exclusion: $n(S \cup D \cup P) = n(S) + n(D) + n(P) - n(S \cap D) - n(S \cap P) - n(D \cap P) + n(S \cap D \cap P)$ $265 = 200 + 110 + 55 - 60 - 30 - n(D \cap P) + 10$ $265 = 285 - n(D \cap P)$ $n(D \cap P) = 285 - 265 = 20$ The number of persons who like only dancing and painting is given by $n(D \cap P) - n(S \cap D \cap P)$. Therefore, $20 - 10 = 10$.

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