14. Six students who share a house all speak exactly two languages. Helga speaks only English and German; Ina speaks only German and Spanish; Jean-Pierre speaks only French and Spanish; Karim speaks only German and French; Lionel speaks only French and English whilst Mary speaks only Spanish and English. If two of the students are chosen at random, what is the probability that they speak a common language?
A $\frac{1}{2}$
B $\frac{2}{3}$
C $\frac{3}{4}$
D $\frac{4}{5}$
E $\frac{5}{6}$
Solution: D
We can select two of the six students in 15 ways, as shown in the table. We have put a tick $(\checkmark)$ in each box corresponding to a pair of students who speak a common language, and a cross ( $\mathbf{x})$ in each box corresponding to a pair of students who do not speak a common language.
We see that there are 12 ticks in the 15 boxes. Therefore, if two students are chosen at random, the probability that they speak a common language is $\frac{12}{15}=\frac{4}{5}$..
Alternative method Note that each of the students has a common language with 4 out of 5 of the other students. It therefore follows that if two students are chosen at random, the probability that they speak a common language is $\frac{4}{5}$.
Note: We may represent the four languages by lines and the students by points which are on the lines corresponding to the languages they speak, as shown. This makes it easy to see that for each student, there is just one other student with whom they have no common language.