(a) When n balls, numbered 1, 2, 3, ... , n are taken in succession from a container, a rencontre occurs if the mth ball withdrawn is numbered m, for some 1 ≤ m ≤ n. Find the probability of getting
(i) No rencontres;
(ii) (Exactly) one rencontre,
(iii) At least one rencontre; and
(iv) r rencontres, where 1 ≤ r ≤ n.
(b) Approximate the answers to the questions in part (a).