- Home
- Standard 11
- Mathematics
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___________.
$10$
$7$
$5$
$11$
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 $10$.