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MTH 4150: Final Practice 1

Problem 1

All faces of cube are painted in \(3\) colors \(R\), \(G\), \(B\). Consider the very specific rotation for \(90^{\circ}\) of the cube around the axis that connects the centers of the top and the bottom face. Let us denote by \(\rho\) this rotation. Determine the number of elements of the set \(\mbox{Inv }(\rho)\).

Problem 2

Determine the number of ways in which the faces of the cube can be painted in \(7\) colors. The paintings that can be obtained from each other using rotation are considered identical.

Problem 3

Find the number of \(7\)-bead necklaces distinct under rotation while under display around a shop mannequin, using three black and four white beads.

Problem 4

Find the number of \(7\)-bead necklaces distinct under rotations or reflections that can be obtained using three black and four white beads.

Problem 5

Assume that \(p\geq 3\) is a prime number. Determine the number of ways in which the vertices of a regular \(p\)-gon can be painted in red and green such that the number of red vertices is odd. Two paintings are considered the same if one can be obtained from the other using rotations and reflections.

Problem 6

Find the number of distinct ways of coloring the faces of a tetrahedron in five colors. Each face has to be colored with one of the colors, and two colorings are considered different if one cannot be obtained from the other using rotations.

Problem 7

Find the number of tables of the form \[\begin{array}{|c|c|}\hline A& B\newline \hline C&D\newline \hline \end{array}\] such that \(A\), \(B\), \(C\), \(D\) are non-negative integers that satisfy \(A+B+C+D=7\). Two tables are considered the same if one can be obtained from the other using rotations or reflections.