In a survey of $100$ agricultural students:
- $60$ study Farm Power ($P$)
- $50$ study Soil Mechanics ($S$)
- $30$ study both Farm Power and Soil Mechanics ($P \cap S$)
Which of the following statements is/are TRUE?
Options:
(A) The number of students studying only Farm Power is $30$
(B) The number of students studying only Soil Mechanics is $20$
(C) The total number of students studying at least one of the two subjects is $80$
(D) The number of students studying neither subject is $20$
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
1. Only $P = n(P) - n(P \cap S) = 60 - 30 = 30$ (A is true).
2. Only $S = n(S) - n(P \cap S) = 50 - 30 = 20$ (B is true).
3. $n(P \cup S) = n(P) + n(S) - n(P \cap S) = 60 + 50 - 30 = 80$ (C is true).
4. Neither $= 100 - n(P \cup S) = 100 - 80 = 20$ (D is true).
All statements A, B, C, D are correct.