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Polya Enumeration Theorem
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Labeling the faces of a cube
Use the Polya Enumeration Theorem with the symmetric group that represents the cube's rotations.
Enumerating the distinct bracelets with 8 beads using 3 colors
Employ the Polya Enumeration Theorem considering the cyclic and dihedral symmetries for bracelets.
Coloring the vertices of a regular hexagon with 2 colors
Use the Polya Enumeration Theorem with the Dihedral group of order 12 (D6) for hexagon symmetries.
Labeling the vertices of a square
Apply the Polya Enumeration Theorem considering the Dihedral group of order 8 (D4) symmetries for the square.
Labeling the edges of a tetrahedron
Use Polya's Theorem, accounting for the tetrahedral group's symmetries.
Coloring the beads of a necklace with 6 beads using 3 colors
Apply Polya's Theorem with the cyclic group of order 6 representing rotations of the necklace.
Labeling diagonals of a regular pentagon using 2 colors
Apply the Polya Enumeration Theorem for the cycle index of the dihedral group D5 acting on the diagonals.
Labeling the sides of a 5-sided pencil
Polya's Theorem is used by considering the symmetry group for a 5-sided prism.
Counting the ways to distribute 4 distinct gifts to 4 children
Use Burnside's Lemma, a corollary of Polya's Theorem, to account for the permutations of the children.
Counting unique arrangements of 4-letter words from alphabet set {A, B, C, D}
Use Polya's Theorem by considering the permutation group on four elements.
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