Question

Use five different colors to paint the four rectangles A, B, C and D shown in the figure. No two rectangles sharing an edge can be the same color. How many ways are there to color the rectangles?

Answer 1

260 ways

Explanation:

There are 3 scenarios.

Scenario #1, all colors used are different. A, B, C, and D each have their own individual color.

In this case, there would be 5x4x3x2=120 ways for this to happen.

(Since each color is different, after one is chosen there are only n-1 colors left, hence why it is 5x4x3x2)

Scenario #2, EITHER A&C OR B&D are the same color (Diagonal rectangles share no edges so they share a color).

In this case, there would be 2(5x4x1x3)=120 ways for this to happen. (Since we have two options, either A&C or B&D share a color, we multiply (5x4x1x3) by two to represent both options. Since the diagonal pairs must share one color, if, for example, A's color has been chosen from 5 options, C has only 1 option to choose from because it MUST match A's color)

Scenario #3, BOTH A&C AND B&D are the same color, meaning only 2 colors are used.

In this case, there would be 5x4x1x1=20 ways for this to happen.

(If A is already one color, B must choose a color from the remaining 4 options, C must match A so it has 1 option, similarily D must match B so it also has only 1 option, hence 5x4x1x1)

Now we add up all possible scenarios.

120+120+20=260 ways