Question

A flag is a 1x3 rectangle made of three 1x1 squares. Each square will be painted in one of 10 possible colors. However, two adjacent squares cannot be painted with the same color. This is the only restriction.

a) How many different flags can be made if the rule given above is followed?

b) How many of these flags use precisely three colors?

c) How many of these use precisely two colors?

Answer 1

a) The number of different flags that can be made if the rule given above is followed is computed here as:

= Number of ways to select one of the 10 colours for the first square * Number of ways to select one of the remaining 9 colours for the middle square * Number of ways to select one of the remaining 9 colours

= 10*9*9

= 810

Therefore 810 different flags are possible here.

b) Number of flags which use exactly 3 colours is computed here as:

= Permutation of 10 colours taken 3 at a time

= 10*9*8

= 720

Therefore 720 flags are possible here.

c) Number of flags with exactly two colours is computed here as:

= Total number of flags possible - Total number of flags with exactly 3 colours

= 810 - 720

= 90

Therefore 90 is the required number of fags possible here.