Modern Math

Set Theory

Medium

Students in a college have to choose at least two subjects from Chemistry, Mathematics and Physics. The number of students choosing all three subjects is $18$ , choosing Mathematics as one of their subjects is $23$ and choosing Physics as one of their subjects is $25$ . The smallest possible number of students who could choose Chemistry as one of their subjects is

$19$

$22$

$20$

$21$

✅ Correct Option: 3

We have students choosing at least 2 subjects from Chemistry (C), Mathematics (M), and Physics (P). This is crucial - no student can choose just one subject.

Given Information:

Students choosing all three subjects = 18

Students choosing Mathematics = 23

Students choosing Physics = 25

Find: Minimum students choosing Chemistry

Since students must choose at least 2 subjects, we have these possible combinations:

Students choosing only M and P (not C) = x

Students choosing only M and C (not P) = y

Students choosing only P and C (not M) = z

Students choosing all three M, P, and C = 18

There are no students choosing only one subject since the constraint requires at least 2 subjects.

From the given totals, we can write:

For Mathematics (M):

Total with M = (M and P only) + (M and C only) + (all three)

23 = x + y + 18

Therefore: $x + y = 5$

For Physics (P):

Total with P = (M and P only) + (P and C only) + (all three)

25 = x + z + 18

Therefore: $x + z = 7$

From our equations:

$x + y = 5$ so $y = 5 - x$

$x + z = 7$ so $z = 7 - x$

For Chemistry (C):

Total with C = (M and C only) + (P and C only) + (all three)

Total with C = $y + z + 18$

Total with C = $(5 - x) + (7 - x) + 18$

Total with C = $30 - 2x$

To minimize the number of students with Chemistry, we need to maximize x.

Constraints on x:

Since $y = 5 - x \geq 0$ , we need $x \leq 5$

Since $z = 7 - x \geq 0$ , we need $x \leq 7$

The limiting constraint is $x \leq 5$ , so maximum $x = 5$ .

When $x = 5$ :

$y = 5 - 5 = 0$

$z = 7 - 5 = 2$

Minimum students with Chemistry = $30 - 2(5) = 20$

Therefore, the smallest possible number of students choosing Chemistry is 20.

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