6

6. [Maximum mark: 14]

The $k$th triangle number, $T_k$, is defined as $T_k = \sum_{r=1}^{k} r$.

(a) (i) Calculate the value of the fifth triangle number, $T_5$.

(ii) Determine the formula for $T_k$ in the form $ak^{2} + bk$. [4]

(b) (i) Find the value of $T_5 + T_4$.

(ii) Find the simplest expression for $T_k + T_{k-1}$. [3]

A bag contains 15 red discs and 10 blue discs, all identical except for colour. Two discs are chosen at random from the bag without replacement.

(c) Calculate the probability that the two discs are different colours. [3]

A bag contains $T_k$ red discs and $T_{k-1}$ blue discs, all identical except for colour. Two discs are chosen at random from the bag without replacement.

(d) Show that the probability that the two discs are different colours is independent of $k$. [4]