Re: IKMC Confusion: Benjamin Paper 2017 Q29

[IbrahimAhmad2004]

Julia has four different coloured pencils and wants to use some or all of them to paint a map of an island divided into four nations. The map of two nations with a common border cannot have the same colour. In how many ways can she colour the map of the map?

[HallsofIvy]

Obviously, she could color all four a different color. There are four ways to choose the first color, three ways to choose the second color, then two ways to choose the third color so there are 4!= 4(3)(2)= 24 ways to do that. Or we could choose 3 areas to color- there are 4!= 12 ways to do that, then color the last the same as the area diagonally opposite. Or choose one color to color two diagonally opposite areas- there are 4 ways to do that, then choose a different color to color the remaining diagonally opposite region. There are 4(3)= 12 ways to do that.

There are 24+ 24+ 12= 60 ways to do this.