In a survey of 220 students of a higher secon | vedclass

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Question

(B) Let $x$ be the number of students who studied all three subjects. Using the Principle of Inclusion-Exclusion for the union of three sets $M, P, C$

Vedclass · Accepted Answer

$45$

Vedclass · Answer

(B) Let $x$ be the number of students who studied all three subjects. Using the Principle of Inclusion-Exclusion for the union of three sets $M, P, C$:$|M \cup P \cup C| = 220 - 10 = 210$.We have $|M \cap P| = 40$,$|P \cap C| = 30$,$|C \cap M| = 50$.The number of students studying only $M$ is $|M| - (40-x) - (50-x) - x = |M| - 90   x$.Similarly,only $P$ is $|P| - (40-x) - (30-x) - x = |P| - 70   x$,and only $C$ is $|C| - (50-x) - (30-x) - x = |C| - 80   x$.The total number of students is:$|M \cup P \cup C| = (|M|   |P|   |C|) - (|M \cap P|   |P \cap C|   |C \cap M|)   |M \cap P \cap C| = 210$.$|M|   |P|   |C| - (40   30   50)   x = 210 \Rightarrow |M|   |P|   |C| = 330 - x$.Given $125 \leq |M| \leq 130$,$85 \leq |P| \leq 95$,$75 \leq |C| \leq 90$.Summing these: $285 \leq |M|   |P|   |C| \leq 315$.Substituting $|M|   |P|   |C| = 330 - x$:$285 \leq 330 - x \leq 315$.$-45 \leq -x \leq -15 \Rightarrow 15 \leq x \leq 45$.Also,from the Venn diagram,the number of students in each region must be non-negative:$40-x \geq 0, 30-x \geq 0, 50-x \geq 0 \Rightarrow x \leq 30$.Combining $15 \leq x \leq 45$ and $x \leq 30$,we get $15 \leq x \leq 30$.Thus,$m = 15$ and $n = 30$.$m   n = 15   30 = 45$.

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