1.2.9. (-) What is the minimum number of trails needed to decompose the Petersen graph? Is there a decomposition into this many trails using only paths?

Write 20 as the direct sum of two of its nontrivial subgroups.

4. The conventional algorithm for evaluating a polynomial anx" + an-1x¹ +... + a1x + ao at x = c can be expressed in pseudocode as Algorithm Polynomial (c, ao, a₁,...,an: real numbers) power = 1; y = ao; for i = 1 to n power power * c; y = y + ai * power return y; Notice that the final value of y is y = anc" + an-1 c¹ +...+ a₁c + ao, the value of the polynomial at x = c. a) Evaluate 3x² + x +1 at x = 2 by working through each step of the algorithm showing the values assigned at each assignment step. b) Exactly how many multiplications and additions are used to evaluate a polynomial of degree n at x = c? (Do not count additions used to increment the loop variable.)

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…