Consider a circular necklace with 18 identical beads. We can rotate the necklace and turn it over. Let G denote the symmetry group.

Suppose we are now given p distinct colors C1, C2, . . . , Cp.

i) Find the number of G-inequivalent ways to color the beads with these colors. Express your answer in terms of p.

ii) Suppose we are given three colors, red and blue and green. Find the generating function for the number of G-inequivalent ways to color the beads red and blue and green.

iii) Find the number of G-inequivalent ways to color the beads, such that 6 beads are red, 6 beads are blue, and 6 beads are green (give the precise numerical answer.)