Two counters of the same colour

A bag contains \(n\) red counters and \(3\) blue counters, where \(n\) is a positive integer. Two counters are taken from the bag at random, without replacement.

1. Show that the probability that the two counters are the same colour is \(\dfrac{n^2-n+6}{(n+2)(n+3)}\).[3]
2. Find the values of \(n\) for which the two counters are equally likely to be the same colour or different colours.[3]
3. Show that the probability that the two counters are different colours can be written as \(\dfrac{6}{n+5+\dfrac{6}{n}}\). Hence show that this probability is never greater than \(\dfrac{3}{5}\), and state the values of \(n\) for which it equals \(\dfrac{3}{5}\).[3]