Some Terms and Notation

1 Name ______Page

CSC 1300 Discrete Structures April 4, 2013

Short Exam 3 of 3

40 minutes, 80 points. No devices with on/off switches, books, notes or other aids may be used. The University’s academic integrity policy governs the conduct of this exam. Credit will be given only for legible work that is clearly associated with a problem statement. Page backs are for scrap work and this work will not be considered part of the answer to a question. Read each question carefully. Ask the instructor if you do not understand the question or the notation.

Some terms and notation: th Fn refers to the n Fibonacci number +++++++++++++++++++++++++++++++++++++

1. (15 points) We have 38 ripe bananas and 8 hungry chimps. How many ways are there to feed the bananas to the chimps if we guarantee that each chimp will get at least 2 bananas?

2. (10 points) The Cookie Factory makes a number of different kinds of cookies, including several varieties with icing, several varieties with nuts (They're carefully labeled respecting the customers with nut alleges), and several varieties with chocolate chips. Right now the display case contains 11 cookies with nuts and chips, 14 with icing and chips, 13 with nuts and icing, and 8 with all three (nuts, chips and icing). What is the smallest possible number of cookies in the display case with icing? 2

3. (10 points) Write out the first five terms of each sequence:

2 a. an = n + n +3 (a0 through a4)

b. an = 2Fn + 4 (a1 through a5)

4. (10 points) Find a closed form for an = an-1 - 6, a0 = 12 3 Name ______Page

5. (20 points) The characteristic equation for an = 4an-1 + 7an-2 - 10an-3 has roots r1 = 1, r2 = -2, r3 = 5 (Please use these root labels in the questions below so we will all generate the same sequences.)

n a. Find initial conditions such that an = (r1)

n n n b. Find initial conditions such that an = (r1) + (r2) + (r3) is the closed form for the recurrence.

c. For each of parts i and ii, use both the recurrence and your closed

form to generate 5 terms (a0 through a4) to show that they are the same. Be sure to show your work carefully. Do not just write down the results. (We know they should be the same!) Show carefully that your results come from both the recurrence relation and the closed form. n c.i. For an = (r1) From closed form: From recurrence formula:

n n n c.ii. an = (r1) + (r2) + (r3) From closed form: From recurrence formula 4

6. (15 points) Find a spanning tree for this graph. Give the degree sequence of the tree you find. (Order of terms counts.)