Answer to Question #248471 in Discrete Mathematics for Justin

Question

At a school education fair, 50 students were interviewed about their interested programmes of study and all of them have shown interest in at least one of the three programmes in the survey. From the results of the interviews, there were 26 students who were interested in studying IT programmes, 18 were interested in Business programmes, 20 were interested in Hospitality programmes and 5 students were interested in all three programmes. How many of the students were interested in exactly

(i) two programmes only?

(ii) one programme only?

Answer 1

Let there be $x$ students who are interested in studying Business and Hospitality programmes only; $y$ students who are interested in Business and IT programmes only; $z$ students who are interested in IT and Hospitality programmes only.

Now we can use Venn diagram:

$50=(13-x-y)+(15-x-z)+(21-y-z)+x+y+z+5$

$50=54-x-y-z$

$x+y+z=4$

So, 4 students are interested in exactly two programmes.

And $50-4-5=41$ students are interested in exactly one programme only.

Answer: i) 4 students; ii) 41 students.