Question

Out of 80 students: $|C|=25$ (Commerce), $|M|=15$ (Mathematics), $|P|=13$ (Physics). Pairwise: $|C\cap M|=3$, $|M\cap P|=4$, $|C\cap P|=2$, and $|C\cap M\cap P|=1$. How many students are studying none of the three subjects?

Hint:

For three sets, memorize \(|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|B\cap C|-|C\cap A|+|A\cap B\cap C|\). “None” is simply total minus the union.

Answer 1

Step 1: Use inclusion--exclusion to find $|C\cup M\cup P|$.
\[ |C\cup M\cup P|=|C|+|M|+|P|-|C\cap M|-|M\cap P|-|C\cap P|+|C\cap M\cap P|. \] Substitute the given numbers: \[ |C\cup M\cup P|=25+15+13-3-4-2+1 = 45. \] Step 2: Subtract from the total to get the count of “none.”
\[ \text{None} = 80 - |C\cup M\cup P| = 80-45=35. \] \[ \boxed{35} \]