Students in a college have to choose at least two subjects

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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