Question

Each of the four walls in a room need to be painted in one color in a way that the adjacent walls are not painted in the same color. If there are three different colors available, in how many ways is it possible to paint the room? It is not necessary to use all colors.

Answer 1

Given : Each of the four walls in a room need to be painted in one color in a way that the adjacent walls are not painted in the same color.

There are three different colors available

To Find :  in how many ways is it possible to paint the room

It is not necessary to use all colors.

Solution:

Case  1 :  Any two colors used ( minimum 2 colors required)

First wall with any 3  colors   ( opposite wall with same color)

and adjacent walls with remaining 2  colors  

Hence 3 * 2=  6 ways  

case 2 :  

All 3 colors used

First wall with 3 colors  

Case 1  : Adjacent wall have  one colors   ( 2 ways)

and opposite wall will  have only 1 option left  

Hence 3 * 2 * 1  = 6 Ways

Case 2  :

Adjacent wall have  2  colors  these 2 colors can be in   2 * 1  =  2  ways

Hence opposite wall need to be same with first wall

Hence 3 * 2  = 6 way  

Total = 6 + 6 + 6  = 18 ways

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