1. State the problem. We are given that 73 students take Sociology, 51 take Demography, 27 take Psychology, 2 take none of the three, 33 take Sociology and Demography, 18 take only Demography, and 5 take Sociology and Psychology. We must find i) total students surveyed ii) number taking only one course iii) number taking at least two courses. 2. Definitions and variables. Let $S$ be the set of students taking Sociology, $D$ Demography, and $P$ Psychology. Let $x$ denote the number taking all three courses, i.e., $x=|S\cap D\cap P|$. 3. Use the Demography total to find the triple intersection. The Demography total equals Demography only plus the regions that also include Demography: $|D|=|D\text{ only}|+|S\cap D\text{ only}|+|D\cap P\text{ only}|+|S\cap D\cap P|$. Plugging the given numbers yields $$51=18+33+|D\cap P\text{ only}|+x$$ This simplifies to $$|D\cap P\text{ only}|+x=0$$ Counts are nonnegative, so $$|D\cap P\text{ only}|=0$$ and $$x=0$$. 4. Compute each region. Sociology only is $$|S\text{ only}|=|S|-|S\cap D\text{ only}|-|S\cap P\text{ only}|-x=73-33-5-0=35$$ Demography only is given as $$|D\text{ only}|=18$$ Psychology only is $$|P\text{ only}|=|P|-|S\cap P\text{ only}|-|D\cap P\text{ only}|-x=27-5-0-0=22$$ The exactly-two regions are $$|S\cap D\text{ only}|=33$$ $$|S\cap P\text{ only}|=5$$ $$|D\cap P\text{ only}|=0$$ The triple intersection is $$x=0$$. 5. Final totals and answers. Total surveyed equals the sum of all seven regions plus none: $$N=35+18+22+33+5+0+0+2=115$$ Number taking only one course is $$35+18+22=75$$ Number taking at least two courses is $$33+5+0+0=38$$ 6. Answers. (i) Total surveyed: $115$. (ii) Only one course: $75$. (iii) At least two courses: $38$.