WAEC 2019 · Paper 2 · Q9

(a)
A survey of 40 students showed that 23 students study Mathematics, 5 study Mathematics and Physics, 8 study Chemistry and Mathematics, 5 study Physics and Chemistry and 3 study all the three subjects. The number of students who study Physics only is twice the number who study Chemistry only. Find the number of students who study: (i) only Physics; (ii) only one subject.
Separate values with commas, e.g. 3, −2
(b)
What is the probability that a student selected at random studies exactly 2 subjects?

Worked solution (try it first)

Draw three overlapping circles, M, P and C, in a rectangle for the 40 students, who each study at least one of the subjects.
Put 3 in the centre.
The "two subjects" numbers include the centre, so the "exactly two" regions are: M and P only $5 - 3 = 2$ , M and C only $8 - 3 = 5$ , P and C only $5 - 3 = 2$ .
Mathematics only $= 23 - 2 - 5 - 3 = 13$ .
Let Chemistry only be $x$ .
Then Physics only is $2x$ .
All the regions add up to 40: $13 + 2 + 5 + 2 + 3 + x + 2x = 40$ , so $25 + 3x = 40$ and $x = 5$ .