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Fifteen balls, including three each of five different colors, are arrange in a triangle as shown. How many ways can this be done if arrangements obtained by rotations are considered the same?

Fifteen balls, including three each of five different colors, are arrange in a triangle as shown. How many ways can this be done if arrangements obtained by rotations are considered the same?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. Fifteen balls, including three each of five different colors, are arrange in a triangle as shown. How many ways can this be done if arrangements obtained by rotations are considered the same? 2. (a) Construct the cycle index polynomial for coloring the faces of the regular tetrahedron. (There are 12 rotations including the identity that map the regular tetrahedron, as a rigid body in 3-space, into itself.) (b) Find the number of ways to paint the faces of a tetrahedron using k colors, for each k = 2, 3, 4. 3. (a) Compute the cycle index polynomial for the group of rotations of the cube represented as permu- tations of the six faces. (There are 24 rotations including the identity that map the cube, as a rigid body in 3-space, into itself. We are ignoring the 24 reflections of the cube.) What is the number of essentially different ways to paint the faces of the cube in 2 colors? In 3 colors? In n colors? (b) What is the number of essentially different ways to paint the faces of a cube such that one face is red, two faces are blue, and the remaining faces are green? Do this 'by hand' (which may be quicker) as well as by using Polya's method, and compare your answers.
Transcribed Image Text:1. Fifteen balls, including three each of five different colors, are arrange in a triangle as shown. How many ways can this be done if arrangements obtained by rotations are considered the same? 2. (a) Construct the cycle index polynomial for coloring the faces of the regular tetrahedron. (There are 12 rotations including the identity that map the regular tetrahedron, as a rigid body in 3-space, into itself.) (b) Find the number of ways to paint the faces of a tetrahedron using k colors, for each k = 2, 3, 4. 3. (a) Compute the cycle index polynomial for the group of rotations of the cube represented as permu- tations of the six faces. (There are 24 rotations including the identity that map the cube, as a rigid body in 3-space, into itself. We are ignoring the 24 reflections of the cube.) What is the number of essentially different ways to paint the faces of the cube in 2 colors? In 3 colors? In n colors? (b) What is the number of essentially different ways to paint the faces of a cube such that one face is red, two faces are blue, and the remaining faces are green? Do this 'by hand' (which may be quicker) as well as by using Polya's method, and compare your answers.
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