Let T = the total number of students

II = # of students who take exactly 2 classes

18 = # of students who take exactly 3 classes

Since students must take at exactly 2 or all 3 classes:

T = II + 13


C = # of students who take chemistry as a class

23 = # of students who take Math

25 = # of students who take Physics
—————————

C + 48 = # tallies of people taking a class

The # of students who take 2 classes will be double counted in the above tallies

The # of students taking 3 classes (18) will be triple counted in the above tallies


Thus


C + 48 = 2 * (II) + 3 * (18)

C + 48 = 2(II) + 54

C = 2(II) + 6


since II stands for the number of people who take exactly 2 classes, it must be an integer.

This means the number of students who take chemistry must be an EVEN number.


20 is the minimum number

If C = 20

20 = 2(II) + 6

14 = 2(II)

II = 7 students who take exactly 2 classes

18 students take exactly 3 classes

This gives us a total of 25 students


To prove the numbers work:


2 can take chemistry and physics; 18 take all 3 ——> 20 take chemistry

0 can take math and chemistry; 5 can take math and physics: and 18 take all three———> 23 take Math

2 take chemistry and physics: 5 take physics and math; and 18 take all three——-> 25 take Physics

The numbers work


20 is the minimum

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all subjects = 18
mathematics
total = 23
so only two subjects where Math is one and other subject is either physics or chemistry is 23-18 =5
physics is 25 so physics being oen and math/chemistry are =7
to minimise chemistry
out of 7 5 be the one whose two subjects are maths and physics are 5
hence remaining 2 are the one whose 2 subjects are chem and physics
thus chem total =2+18= 20
C
_________________

You can use Venn diagram to get to the answer.

_________________

A 22
B 19
C 20
D 21
E 23
All students who chose 3 class = 18
Mandatory 2 classes therefore the students who cannot chose between Maths and physics = 25-23 =2

Therefore the total no of students who chose Chemistry = 18+2 = 20

Therefore IMO A

Answer should be A

IF WE SUBTRACT 18 FROM 23 AND 25...THEN WE LEFT WITH 5 AND 7...

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