1. Problem statement: In a survey students were asked if they take demography D, sociology S, and psychology P with counts $n(S)=73$, $n(D)=51$, $n(P)=27$, $n(S∩D)=33$, $n(S∩P)=5$, only demography $=18$, and none $=2$. 2. Let $x$ denote the number taking all three courses, i.e., $x=n(S∩D∩P)$. 3. Use the formula for only demography: only demography $=n(D)-n(S∩D)-n(D∩P)+n(S∩D∩P)$. 4. Substitute known values to get $$18 = 51 - 33 - n(D∩P) + x$$ 5. Solve for $n(D∩P)$ to find $n(D∩P)=x$. 6. Therefore the region D∩P excluding S is $n(D∩P)-x=0$. 7. Compute the "only" regions and pairwise-only regions as follows. 8. Only S $=n(S)-n(S∩D)-n(S∩P)+x = 73 - 33 - 5 + x = 35 + x$. 9. Only P $=n(P)-n(S∩P)-n(D∩P)+x = 27 - 5 - x + x = 22$. 10. SD only $=33 - x$, SP only $=5 - x$, DP only $=0$. 11. Total surveyed $N$ is the sum of all seven regions plus none, so $$N = (35 + x) + 18 + 22 + (33 - x) + (5 - x) + 0 + x + 2$$ 12. Simplifying gives $N = 115$. 13. Number taking only one course is only S + only D + only P $= (35 + x) + 18 + 22 = 75 + x$. 14. Number taking at least two courses is SD only + SP only + DP only + triple $= (33 - x) + (5 - x) + 0 + x = 38 - x$. 15. Constraint: $x$ is an integer with $0 ≤ x ≤ 5$ since SP only $=5 - x ≥ 0$ and counts are nonnegative. 16. Final answers: i) $115$ surveyed. 17. ii) Number taking only one course $=75 + x$, which ranges from $75$ to $80$ depending on $x$. 18. iii) Number taking at least two courses $=38 - x$, which ranges from $33$ to $38$ depending on $x$.