French Flag Colors

National Flag of France has 3 color stripes: red,white, blue = (r,w,b)
If we permute the 3 colors, how many flags are there ?

Eg. (w,b,r) = (r,b,w) since they are same if flip over the flag.
Eg. (b,b,b),(r,w,r)... repeat color allowed.
[Ans: 18]
Hint: Apply Group action on the Set (r,w,b)


French Flag:
Let Group G = {g1,g2 }
g1={e}= {(1)(2)(3)}
g2={(1 3)(2)}
|G|=order = 2
Apply Group Action on 3 colors set X {r,w,b}:
g1.X=e.{r,w,b} => Fix 3 stripes each with 3 colors
=> |Fix(g1)|= 3³ ways

(1 3)= Fix 1st & 3rd stripe as 1 stripe, x 3 colors => 3 ways
(2) => Fix 2nd stripe x 3 colors = 3 ways

g2.X=(1 3)(2).X=> 3x3
=> |Fix(g2)|= 3²

By Counting Theorem:
Total Flags = Orbit(X)
= ∑|Fix(g) | /|G|
= (3³ + 3² ) / 2
= 18



Exercise:
Chessboard (8x8=64) colored arbitrarily in black or white.
How many different patterns are there ?

[Consider 2 patterns as the same if a rotation (90, 180, 270 degrees) takes one to another ?]