Question

A First Course in Abstract Algebra (7th Edition)

Chapter S.17, Problem 4E

Wooden cubes of the same size are to be painted a different color on each face to make chil- dren's blocks. How many distinguishable blocks can be made if 8 colors or paint are available?

Hint: X must be a set of functions from a set with 6 elements to a set with 8 elements. Describe it carefully. Then explain how G acts on X.

I konw that A= (1,2,3,4,5,6) which corresponds to sides in the cube and B=(1,2,3,4,5,6,7,8) which corresponds to painting on the sides where F: Aـــــــــــ>B. But I do not know how to reach a solution. Can I get a solution through you with thanks?

Answer 1

For painting any one of the face you have 8 choices of colours available with you.

Now for painting second face you have only 7 choices of colour available with you because of the eight colours available wwas utilised for paunting the first face.

Similarly for painting third face you have 6 vhoices of colour available with you.

And so on for the last face you have 3 choices of colour available with you.

Therefore total ways in which you can paint the cube is

8×7×6×5×4×3 = 20160

Hence 20160 blocks are available with you.