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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

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 $n = 30$.$m   n = 15   30 = 45$.

Similar Questions

In a battle,$70\%$ of the combatants lost one eye,$80\%$ an ear,$75\%$ an arm,$85\%$ a leg,and $x\%$ lost all the four limbs. The minimum value of $x$ is:

Let $S = \{(x, y, z) : x, y, z \in \mathbb{Z}, x + 2y + 3z = 42, x, y, z \geq 0\}$. Find the number of elements in $S$.

In a class of $60$ students,$30$ opted for $NCC$,$32$ opted for $NSS$,and $24$ opted for both $NCC$ and $NSS$. If one of these students is selected at random,find the probability that the student opted for $NCC$ or $NSS$.

If $X$ and $Y$ are two sets such that $X$ has $40$ elements,$X \cup Y$ has $60$ elements and $X \cap Y$ has $10$ elements,how many elements does $Y$ have?

Twenty persons arrive in a town having $3$ hotels $x, y$ and $z$. If each person randomly chooses one of these hotels,then what is the probability that at least $2$ of them go to hotel $x$,at least $1$ to hotel $y$,and at least $1$ to hotel $z$? (Each hotel has a capacity for more than $20$ guests.)

Vedclass Test Series

Exam Paper Generator

Online Exam Module