Question 9

Let $D, C, M$ denote the sets of students in Drama, Dance, and Maths clubs, respectively. We are given:

• Only Drama: $|D \setminus (C \cup M)| = 60$
• Only Dance: $|C \setminus (D \cup M)| = 80$
• Only Maths: $|M \setminus (D \cup C)| = 30$
• In all three clubs: $|D \cap C \cap M| = 2$

We are also given the two-club overlaps:

• Both Drama and Dance: $|D \cap C| = 40$
  Thus, the number of students in Drama and Dance only is $40 - 2 = 38$.
• Both Dance and Maths: $|C \cap M| = 12$
  Thus, the number of students in Dance and Maths only is $12 - 2 = 10$.
• Both Drama and Maths: $|D \cap M| = 7$
  Thus, the number of students in Drama and Maths only is $7 - 2 = 5$.

The total number of students in at least one club is the sum of all disjoint regions:

We are given that $75\%$ of students in the college are not in any of these clubs. Thus, the $225$ students represent $100\% - 75\% = 25\%$ of the total student body $T$: