Assignment:
Q1. Use the theorem below to determine the number of nonequivalent colorings of the corners of a rectangle that is not a square with the colors red and blue. Do the same with p colors.
Theorem: Let G be a group of permutations of X and let C be a set of colorings of X such that f * c is in C for all f in G and all c in C. Then the number N(G,C) of nonequivalent colorings in C is given by
N(G,C) = (1/absG)Σ(abs(C(f)), (in words the number of nonequivalent colorings in C equals the average of the number of colorings fixed by the permutations in G)
Q2. A two-sided marked domino is a piece consisting of two squares joined along an edge where each square on both sides of the piece is marked with 0,1,2,3,4,5 or 6 dots.
a. use the theorem above to determine the number of different two sided marked dominoes.
b. how many different two sided marked dominoes are there if we are allowed to mark the squares with 0,1,...p-1 or p dots?
Provide complete and step by step solution for the question and show calculations and use formulas.