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

The correct answer is 20.

Let's carefully set up this problem. Each student chooses at least two subjects from Chemistry (C), Mathematics (M), and Physics (P). This means there are exactly four possible combinations a student can be in:

• Only M and P (not C) — let this count be a
• Only M and C (not P) — let this count be b
• Only P and C (not M) — let this count be c
• All three M, P, and C — given as d = 18

Now, let's translate the given information into equations.

Step 1: Use the Mathematics count.

Students choosing Mathematics include those in (M and P only), (M and C only), and (all three):

$$a+b+d=23$$

Since d = 18:

$$a+b=23-18=5$$

Step 2: Use the Physics count.

Students choosing Physics include those in (M and P only), (P and C only), and (all three):

$$a+c+d=25$$

Since d = 18:

$$a+c=25-18=7$$

Step 3: Write the Chemistry count.

Students choosing Chemistry include those in (M and C only), (P and C only), and (all three):

$$\text{Chemistry count}=b+c+d=b+c+18$$

To minimize the Chemistry count, we need to minimize $b+c$.

Step 4: Express b and c in terms of a.

From Step 1: $b=5-a$

From Step 2: $c=7-a$

Therefore:

$$b+c=(5-a)+(7-a)=12-2a$$

This is minimized when a is as large as possible.

Step 5: Find the maximum value of a.

Since all counts must be non-negative integers:

• $b=5-a\ge 0$  ⇒  $a\le 5$
• $c=7-a\ge 0$  ⇒  $a\le 7$

The binding constraint is $a\le 5$, so the maximum value of a is 5.

Step 6: Calculate the minimum Chemistry count.

Setting a = 5:

$$b=5-5=0,c=7-5=2$$

$$\text{Minimum Chemistry count}=0+2+18=20$$

Verification: With a = 5, b = 0, c = 2, d = 18:
• M count: 5 + 0 + 18 = 23 ✓
• P count: 5 + 2 + 18 = 25 ✓
• All three: 18 ✓
• Every student is choosing at least two subjects ✓

Therefore, the smallest possible number of students who could choose Chemistry as one of their subjects is 20.