Answer to Question #279379 in Discrete Mathematics for Jaishree

Question

Let A, B, C, D denote, respectively, art, biology, chemistry, and drama courses.

Find the number N of students in a dormitory given the data:

12 take A, 5 takeAand B, 4 takeB and D, 2 take B, C,D,

20 take B, 7 takeAand C, 3 takeC and D, 3 take A, C,D,

20 take C, 4 takeAand D, 3 take A, B,C, 2 take all four,

8 take D, 16 takeB and C, 2 take A, B, D, 71 take none.

Answer 1

Let T be the number of students who take at least one course. By the Inclusion-Exclusion Principle Theorem, $T=s_{1}-s_{2}+s_{3}-s_{4}$ where:

$\begin{array}{ll} s_{1}=12+20+20+8=60, & s_{2}=5+7+4+16+4+3=39, \\ s_{3}=3+2+2+3=10, & s_{4}=2 . \end{array}$

Thus T=29, and N=71+T=100.