24. Part of a wall is to be decorated with a row of four square tiles. Three different colours of tiles are available and there are at least two tiles of each colour available. Tiles of all three colours must be used.
In how many ways can the row of four tiles be chosen?

A 12
B 18
C 24
D 36
E 48
24. D First note that as there are four tiles to be placed and all three colours must be used, every arrangement of tiles consists of two of one colour and one each of the other two colours. Let the colours be R, G and B and consider the arrangements in which there are two tiles of colour R. These two tiles may be placed in six different ways: $R R * *, R * R *, R * * R, * R R *, * R * R$ and $* * R R$. For each of these arrangements of $R$ tiles, there are two possible ways of placing the remaining $\mathrm{G}$ tile and $\mathrm{B}$ tile - the $\mathrm{G}$ tile may go in the first remaining space or the second remaining space and then there remains only one space for the $\mathrm{B}$ tile. So the number of arrangements in which there are two $\mathrm{R}$ tiles is $2 \times 6=12$. By the same reasoning, we see that there are 12 different arrangements in which there are two $\mathrm{G}$ tiles and 12 different arrangements in which there are two $\mathrm{B}$ tiles. So the total number of different arrangements is $3 \times 12=36$.