Please solve these questions

10. 11. 12. 13. \"um-u;- uu-l...u uu1. _;; u;_ .m...u-..;u., . Suppose that in Exercise 4 we permit only NH2 and COOH for the radicals. How many compounds are possible? . Determine the number of ways in which the faces of a regular dodecahedron (regular lZsided solid) can be colored with three colors. . Determine the number of ways in which the edges of a square can be colored with six colors so that no color is used on more than one edge. . Determine the number of ways in which the edges of a square can be colored with six colors with no restriction placed on the number of times a color can be used. . Determine the number of different llbead necklaces that can be made using two colors. Determine the number of ways in which the faces of a cube can be colored with three colors. Suppose a cake is cut into six identical pieces. How many ways can we color the cake with :1 colors assuming that each piece receives one color? How many ways can the ve points of a vepointed crown be painted if three colors of paint are available? Let G be a nite group and let sym(G) be the group of all permuta tions on G. For each g in G, let gbg denote the element of sym(G) de ned by g(x)= gxg'l for all x in 8G. Show that G acts on itself under the action g > qbg. Give an example 1n which the mapping g > gbg 1s not onetoone. 1. Find a Hamiltonian circuit in the digraph given in Example 7 dif- ferent from the one in Figure 30.3. 2. Find a Hamiltonian circuit in Cay({ (a, 0), (b, 0), (e, 1 ) } :2 072). 3. Find a Hamiltonian circuit in Cay({(a, 0), (b, 0), (e, 1)}:24 0Zm) where m is even. 4. Write the sequence of generators for each of the circuits found in Exercises 1, 2, and 3. 5. Use the Cayley digraph in Example 7 to evaluate the product a ba-'bab-1