$\begin{align}  & \begin{array}{*{35}{l}}   Let\text{ }M\text{ }be\text{ }the\text{ }set\text{ }of\text{ }those\text{ }who\text{ }offer\text{ }Mathematics  \\   Let\text{ }P\text{ }be\text{ }the\text{ }set\text{ }of\text{ }those\text{ }who\text{ }offer\text{ }Physics  \\\end{array} \\ & n(\xi )=40 \\ & n(M)=32 \\ & n(P)=24 \\ & n{{(M\cup P)}^{1}}=4 \\ & n(M\cap P)=x \\ & n(M\cap {{P}^{1}})=32-x \\ & n({{M}^{1}}\cap P)=24-x \\ & 40=(32-x)+x+(24-x)+4 \\ & 40=60-x \\ & x=20;\text{   }n(M\cap P)=20 \\\end{align}$