Take-Home Exam #1
Due Thursday, October 8

You may refer to your class notes and ask me if you have questions, but you are not permitted to use any other source of help or discussion, whether human or nonhuman.

1. Do problem #4 of Section 2.7 by induction.
2. Finish the problem that we began in class: How many different tetrahedra can be produced by coloring each face a solid color and using n different colors? (Two tetrahedra are the same if they can be turned and placed side by side so that corresponding sides match in color.) So far, the numbers for n=0,1,2,3,4 are 0,1,5,15,36. Find and prove a general formula.
3. 1. Use finite differences to guess a formula for the sum of the fourth powers of the first n natural numbers.
   2. Work problem #4 of Section 3.8. However, there is a correction. The final formula should read:

      

      Then use the formulas in problems #1, 5, and 6 of Section 2.7 for , , and to derive the formula for . Verify that you got the same answer as in (a).

4. Just to be certain everything is clear, give a concise explanation for the formula for the number of injective functions from X to Y if |X|<|Y|.
5. Work problem #14 of Section 4.4. Suggestion: Consider the oddness/evenness of each coordinate.
6. Work problem #5 of Section 4.5.
7. Work problem #9 of Section 4.5.
8. Work problem #10 of Section 4.5.

• About this document ...