Final Exam Review Answers

I did not prove most of these things. I would like you to try to do this on your own.

1. Pigeonhole Principle: Know how to solve basic problems and be able to state what the principle says (precisely). • There are six math focus classes offered at WOU. What would be the least number of people needed to sign up for classes to ensure that there are 8 people in at least one class? 43 • We have 10 boxes labeled 1 through 10 into which we place pennies. How many pennies are required to ensure that at least one box contains at least as many pennies as the label on the box? 46

2. Venn Diagrams: Know how to use the Principle of Inclusion-Exclusion and be able to state what the principle says (precisely). • How many integers between 1 and 450 (inclusive) are not divisible by 2, 5, or 7? 154 • How many integers between 0 and 9,999 (inclusive) have among their digits each of 2, 5, and 8? 204

3. Gauss Summing: • Find the sum of the multiples of 3 that are less than 600 and are not divisible by 4. 45000 • Find the sum of the 1st 400 positive multiples of 4. 320800

4. Permutations/Combinations: Explain how to generate the formula for each. As well as: ⎛n⎞ • P(n ,n) = n! ⎜ ⎟ = n ⎝1⎠ 5. Suppose you want to seat five boys and five girls along one side of a long table with 10 seats. • If the boys and girls are distinguishable, how many ways are there to seat these ten children? 10! • If the boys and girls are indistinguishable (This means we don’t know names. We just know B and G.), how many ways are there to seat these ten children? ⎛10⎞ ⎜ ⎟ ⎝ 5 ⎠ • If the boys and girls are distinguishable and the boys and girls alternate seats, how many ways are there to seat these ten children? 2(5!)2

6. Relate combinations to Pascal’s Triangle. If you want to make a pizza with 5 toppings ⎛8⎞ and you have 8 toppings available, how many pizzas can you make? ⎜ ⎟ ⎝5⎠

7. : Define and draw an example of each term. Also, tell me what you know about each term: • Kn ● Cn (This is a of length n) • Planar graphs ● Graph with an Euler circuit / trail

8. Basic Graph Theory Questions: n(n −1) • How many edges in Kn? 2 • Prove that the sum of the degrees in a graph is equal to twice the number of edges in the graph. Each edge contributes one to the of each of its endpoints, i.e., adds two to the sum of the degrees.

9. More Graph Theory Questions: • Can a graph have an odd number of odd degree vertices? No. • Can a bipartite graph have an odd cycle? No. A cycle starts on (say) the left and ends on the left. This requires an even number of edges, so it requires an even number of vertices (because it is a cycle).

• Show that the number of vertices in a k-regular graph is even if k is odd.

A graph on n vertices that is k-regular has kn/2 edges (because the sum of the degrees is kn = 2*# of edges). If k is odd, then n has to be even in order for that fraction kn/2 to be an integer.

10. What is Euler’s formula for planar graphs? V-E+F=2