Colourful Tiles
Weekly Problem 21 - 2011
How many ways can you paint this wall with four different colours?
How many ways can you paint this wall with four different colours?
Age
11 to 14
Challenge level
A tile is fixed to a wall and then painted with four different colours, one for each quarter. One way of doing this is
Image
How many other ways can the tile be painted?
Consider the colour at the top. There are $4$ different choices for this: red, yellow, green and blue. For the right hand colour, this can be any of the other $3$ colours that have not yet been used. The bottom colour can then be either of the other $2$ colours. The left hand part then has to have the remaining colour.
This means there are $4 \times 3 \times 2 \times 1 = 24$ different ways to paint all four sections. This means there are $23$ other ways to paint the tile.