In a batch of $100$ agricultural engineering postgraduate students:
- $50$ students study Farm Power & Machinery ($M$)
- $45$ students study Soil & Water Conservation ($S$)
- $40$ students study Food Process Engineering ($F$)
- $20$ students study both $M$ and $S$
- $15$ students study both $S$ and $F$
- $18$ students study both $M$ and $F$
- $8$ students study all three subjects ($M, S, F$)
Which of the following statements is/are TRUE?
Options:
(A) The number of students studying at least one of the three subjects is $90$.
(B) The number of students who do not study any of the three subjects is $10$.
(C) The number of students studying exactly one subject is $53$.
(D) The number of students studying exactly two subjects is $35$.
Answer Key:
A, B, C
Step-by-Step Solution & Explanation
**Method 1: Three-Set Venn Inclusion-Exclusion**
1. At least one subject:
$|M \cup S \cup F| = (50 + 45 + 40) - (20 + 15 + 18) + 8 = 135 - 53 + 8 = 90$
Statement A is True.
2. Neither subject: $100 - 90 = 10$. Statement B is True.
3. Exactly two subjects:
$(20 - 8) + (15 - 8) + (18 - 8) = 12 + 7 + 10 = 29$
Statement D claims 35, which is False.
4. Exactly one subject:
- Only $M = 50 - (12 + 10 + 8) = 20$
- Only $S = 45 - (12 + 7 + 8) = 18$
- Only $F = 40 - (10 + 7 + 8) = 15$
$\text{Sum} = 20 + 18 + 15 = 53$
Statement C is True.
Therefore, options **A**, **B**, and **C** are correct.