A (one-sided) marked domino isa piece consisting of two squares joined along an edge, where eachsq)lare on one side of the piece is marked with 0, 1,2,3,4,5, or 6dots. The two squares of a marked domino may receive the samenumber of dots.

(a) Use Theorem 14.2.3 to determine the number of differentmarked dominoes.

(b) How many different marked dominoes are there if we areallowed to mark the squares with 0,1, ... ,p - 1, or p dots?

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Let X be a set of n elements, and suppose we have a set of k colors available with which to color the elements of X. Let C be the set of all k" colorings of X. Let G be a group of permutations of X. Then the number of nonequivalent colorings is the number N(G,C) = PG(k, k,..., k) obtained by substituting z₁k, (i=1,2,...,n) into the cycle index of G. Proof. This theorem is a consequence of Theorems 14.2.3 and 14.3.1. The cycle index of G is the average PG(21,22,..., Zn) 14.3. POLYA'S COUNTING FORMULA of the sum of the monomials associated with the permutations f in G. By Theorem 14.3.1, the number of colorings in C that are fixed by f equals k#(1) kestest ten = = where (e1,e2,..., en) is the type of f. By Theorem 14.2.3, the number of nonequivalent colorings is 1 N(G,C) GΣkkea ken = Pa (k,k, ..., k). JEG Let X be a set of n elements, and suppose we have a set of k colors available with which to color the elements of X. Let C be the set of all k" colorings of X. Let G be a group of permutations of X. Then the number of nonequivalent colorings is the number N(G,C) = PG(k, k,..., k) obtained by substituting z₁k, (i=1,2,...,n) into the cycle index of G. Proof. This theorem is a consequence of Theorems 14.2.3 and 14.3.1. The cycle index of G is the average PG(21,22,..., Zn) 14.3. POLYA'S COUNTING FORMULA of the sum of the monomials associated with the permutations f in G. By Theorem 14.3.1, the number of colorings in C that are fixed by f equals k#(1) kestest ten = = where (e1,e2,..., en) is the type of f. By Theorem 14.2.3, the number of nonequivalent colorings is 1 N(G,C) GΣkkea ken = Pa (k,k, ..., k). JEG