Question

There are 32 students in a class. Three courses namely English, Hindi and Mathematics are offered to them. Each student must register for at least one course. If 16 students take English, 8 students take Hindi, 18 students take Mathematics, 4 students take both English and Hindi, 5 students take both Hindi and Mathematics, and 5 students take both English and Mathematics, then the number of students who take Mathematics only is (in integer).

Answer 1

Correct Answer: 12

\[ \text{Let: } E = \text{English}, \quad H = \text{Hindi}, \quad M = \text{Mathematics}. \] \[ \text{Given values:} \] \[ |E \cap H| = 4, \quad |H \cap M| = 5, \quad |E \cap M| = 5, \] \[ |E| = 16, \quad |H| = 8, \quad |M| = 18. \] \[ \text{We need to find the number of students who take only Mathematics, } |M \setminus (E \cup H)|. \] \[ \text{Using the principle of inclusion-exclusion:} \] \[ |E \cup H \cup M| = |E| + |H| + |M| - |E \cap H| - |H \cap M| - |E \cap M| + |E \cap H \cap M|. \] \[ \text{Substituting the given values:} \] \[ |E \cup H \cup M| = 16 + 8 + 18 - 4 - 5 - 5 + |E \cap H \cap M|. \] \[ = 28 + x, \quad \text{where } x = |E \cap H \cap M| \text{ (students taking all three subjects)}. \] \[ \text{Since } |E \cup H \cup M| \leq 18, \text{ the number of students taking only Mathematics is:} \] \[ |M \setminus (E \cup H)| = |M| - (|E \cap M| + |H \cap M| - |E \cap H \cap M|). \] \[ \text{Substituting the values:} \] \[ |M \setminus (E \cup H)| = 18 - (5 + 5 - x). \] \[ = 18 - 10 + x = 8 + x. \] \[ \text{If } x = 4 \text{ (students taking all three subjects):} \] \[ |M \setminus (E \cup H)| = 8 + 4 = 12. \] \[ \text{Final Answer: } \mathbf{12}. \]