The mathematics course at the University of Barchester is a three-year one. After the end-of-year examinations there are three possibilities:

(i) failing and leaving (probability $p$ );

(ii) taking that year again (probability $q$ );

(iii) going on to the next year (or graduating, if the current year is the third one) (probability $r$ ).

Thus there are five states for a student $\left(1^{\text {st }}\right.$ year, $2^{\text {nd }}$year, $3^{\text {rd }}$year, left without a degree, graduated).

Write down the $5 \times 5$ transition matrix. Classify the states, assuming $p, q, r \in(0,1)$. Find the probability that a student will eventually graduate.