A group of 40 students appeared in an examination of 3 subjects - Mathematics, Physics Chemist

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1.Set Theory

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A group of $40$ students appeared in an examination of $3$ subjects - Mathematics, Physics Chemistry. It was found that all students passed in at least one of the subjects, $20$ students passed in Mathematics, $25$ students passed in Physics, $16$ students passed in Chemistry, at most $11$ students passed in both Mathematics and Physics, at most $15$ students passed in both Physics and Chemistry, at most $15$ students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is___________.

A

$10$

B

$7$

C

$5$

D

$11$

(JEE MAIN-2024)

Solution

$11-x \geq 0$ Maths and Physics

$\mathrm{x} \leq 11$

$\mathrm{x}=11$ does not satisfy the data.

$ 11+z \leq 15 \Rightarrow z \leq 4$

$ 11+y \leq 15 \Rightarrow y \leq 4$

Now

$ 9-z+0+14-y+z+11+y+5-y-z=40$

$ \Rightarrow y+z=-1$

Not possible

$\Rightarrow \mathrm{x} \leq 10$

For $\mathrm{x}=10$

Hence maximum number of students passed in all the three subjects is $10$.

Standard 11

Mathematics

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