Higher June 2023 Paper 3 Q24

24 There is a total of \(y\) counters in a box.

There are \(x\) pink counters and 5 blue counters in the box.
The rest of the counters are green.

\(x : y = 1 : 3\)

Freda takes at random two counters from the box.

Find, in terms of \(x\), an expression for the probability that Freda takes two counters of the same colour.

Give your answer as a fraction in the form \(\dfrac{ax^2 + bx + c}{dx^2 + ex}\) where \(a\), \(b\), \(c\), \(d\) and \(e\) are integers. (5)