Final Exam from 2007 Solutions
1. Suppose that we have 10, 18, and 28 stamps, each in unlimited supply. Let s(n) be the number of ways
of obtaining n cents of postage if the order in which we put on stamps counts. For example, s(10) = 1 and
s(20) = 1 (two
Homework Week 7 Solutions
pp. 261 2
#30 Note that the tiling of length n, either begins with a white, blue or red.
If it begins with a white, there are hn1 ways to tile the rest.
If it begins with a blue, then we need n 1 more tiles, where an even number
Homework Number 8 Solutions
HW pp. 449 450:
#3 No because the graph has an odd number of odd-degree vertices (namely one).
#4 No. Since 3 of the 5 vertices have maximum degree (4), this means these three vertices are
adjacent to each vertex. Therefore, th
Homework Week 6 Solutions
pp. 258 261
#11a. Proof by induction on n.
Base Case (n = 1): LHS = l1 = 1
RHS = f0 + f2 = 0 + 2 = 1
Induction Hypothesis: Assume lk = fk1 + fk+1 for all k < n.
Induction Step: Prove this is true for n, i.e. ln = fn1 + fn+1 .
LHS
Homework Week 5 Solutions
pp. 199
#15a. D7
#15b. 7! D7 (got rid of the derangements)
#15c. 7! D7 7 D6 (got rid of the derangements and the ones with exactly one right.and the
other six deranged)
#16 LHS = number of ways to arrange n objects. Each of these
Homework Week 2 Solutions
1. How many license plates involving one, two, or three letters and one, two, or three digits are there if
the letters must appear in a consecutive grouping?
Since the letters have to appear in a consecutive grouping, we can thin
Homework Week 3 Solutions
1. In the daily Play 4, players select a number between 0000 and 9999. The lottery machine
contains 4 bins, each with 10 ping-pong balls. Each bin has its ping-pong balls labeled
0, 1, 2, . . . , 9. The state then randomly select
Homework Week 4 Solutions
pp. 154 160 :
18 5
3 (2)13 . The coecient of x8 y 9 is 0 since the degree of the
5
polynomial is 18 and the degree of x8 y 9 is 17.
#6 The coecient of x5 y 13 is
#11 Let a, b, c be special elements of an n-set (so there are n 3 o
HW Week 3
Due Wednesday October 21
1. In the daily Play 4, players select a number between 0000 and 9999. The lottery machine
contains 4 bins, each with 10 ping-pong balls. Each bin has its ping-pong balls labeled
0, 1, 2, . . . , 9. The state then random