Sample questions for the midterm:

1. How many bitstrings of length 10 have exaclty four zeros?
2. What is the coefficient of x^3*y^6*z^5 in (x+y+z)^14? Explain in
words why your answer is correct.
3. How many words of length 7 contain both `a' and `b'?
Solutions to Assignment 8
8.2:
4.d: chi(r)=(r1)^2. So a_n = A*2^n + B*n*2^n. Initial conditions show
A=4, B=7/2.
4.e: chi(r)=(r1)(r+1). So a_n=A+B*(1)^n. Initial conditions give
A=2, B=3.
4.f: chi(r)=(r+3)^2. So a_n=A*(3)^n + B*n*(3)^n. Initial c
Extra Problem 1:
Suppose 2n people sit on a round table and are shaking hands in
pairs. Suppose that etiquette is observed and no 2 shakes cross. Let
S_n be the number of possible shaking hands arrangements of this sort.
Determine S_10.

Extra Problem
Math 373
Test 1
Fall 2012
September 27, 2012
1. Meng takes out a loan to buy a new motorcycle. The amount of the loan is 12,500. Meng will
repay the loan with 9 monthly payments of Q at a nominal interest rate of 12% compounded
monthly.
Calculate Q.
Solut
MATH 373
Test 3
Fall 2012
November 6, 2012
1. The preferred stock of Wang Corporation pays a quarterly dividend of 5.00. The next
dividend is paid in 1 month.
Yi buys the stock to yield an annual effective rate of 10%.
Using the dividend discount method,
Math 373
Test 3
Fall 2013
November 7, 2013
1. You are given the following spot interest rate curve:
Time t
Spot Rate rt
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
3.2%
3.5%
3.9%
4.4%
5.0%
5.7%
6.5%
7.5%
Calculate the accumulated value of a three year annuity immedia
Math 373
Test 2
Fall 2013
October 17, 2013
1. You are given the following table of interest rates:
2007
2008
2009
2010
2011
2012
2013
Year 1
0.060
0.055
0.053
0.050
0.047
0.045
0.043
Year 2
0.058
0.052
0.051
0.049
0.046
0.044
Year 3
0.056
0.049
0.048
0.04
Math 373
Spring 2012
Test 2
March 8, 2012
1. A 20 year callable bond with semiannual coupons of 250 matures for 11,000. The bond can be
called at the end of 12, 14, 16, and 18 years. The following call values apply:
End of Year
12
14
16
18
Call Value
11,
@A’. A 20 year callable bond with semiannual coupons of 250 matures for 11,000. The bond can be
called at the end of 12, 14, 16, and 18 years.
End of Year
Call Value
The following call values apply:
Determine the price that Kendrick should pay to a
Math 373
Test 3
Spring 2012
April 3, 2012
1. A stock pays quarterly dividends with the next dividend payable in one month. The first
dividend will be 4.00. The second dividend will be 4.50. The third dividend will be 5.00. The
dividends are expected to co
Math 373
Test 3
Spring 2012
April 3, 2012
1. A stock pays quarterly dividends with the next dividend payable in one month. The first
dividend will be 4.00 . The second dividend will be 4.50 . The third dividend will be 5.00 . The
dividends are expected to
Math 373
Spring 2013
Test 1
February 12, 2013
1. Tracy is receiving an annuity immediate with quarterly payments of 250 for 10 years. Tracy
invests each payment in an account earning an annual effective interest rate of 8%.
Determine the amount the Tracy
MATH 373
Test 3
Spring 2013
April 2, 2013
1. The stock of Hillman Industries pays quarterly dividends. The next dividend will be D and
is payable in two months.
Future dividends are expected to increase each quarter with each dividend being 101.5% of
the
Math 373
Test 2
Spring 2013
March 5, 2013
1. Jana purchased a 20 year zero coupon bond for 20,000. The bond matures for 70,000.
Christian borrowed 50,000 to be repaid over 30 years with level monthly payments.
, The annual effective interest rate on Chri
Math 373
Fall 2012
Test 2
October 18, 2012
1. Jordan has the option to purchase either of the two bonds below. Both bonds will be purchased
to provide the same yield rate.
a. A 20year zero coupon bond which matures for 100,000 and has a price of 50,000.
Sample questions for the 375 Final
* 11.1: 28,31,47  11.4: 1,9,33,39,44  11.5: 8.
* Give a recurrence and initial condition for the number of ternary
strings that contain 3 or more consecutive zeros.
* Solve a_n = 5*a_cfw_n1  4*a_cfw_n2 +3*2^n with
Assignment 1
MA 375 Homework 1 Solutions
Section 5.1:
54: The claim is true if n=1 because if you pick 2 numbers from the
set cfw_1,2 then it is easy to see that there are 2 chosen
numbers that divide each other. (Generally, if 2 equal numbers
are cho
Section 4.2
29: By "strong" induction. If n=1, it's clear that n can be uniquely
written as a sum of distinct powers of 2, namely n=2^0.
For n>1, subtract from n the largest power of 2 not exceeding it:
n'=n2^a. Observation: 2n'<n. Indeed, if 2n'n=2n2*
Solutions 3
7.1:
19: First find the number of hands with 5 different kinds: we choose 5
kinds out of the possible 13, and then each kind can be any of 4
suits. So there are (13 choose 5) * 4^5 such hands.
In class we found how many flushes there are:
Solutions Assignment 4:
7.2:
2: If 3 is twice as likely as all other numbers,
p(3)/2=p(1)=p(2)=p(4)=p(5)=p(6). Let x=p(1).
Since the sum of all p(i) is 1, x+x+2x+x+x+x+x=1, so x=1/7.
10: a) Note that the 13th letter is "m", and there are 26! ways of
Assignment 5
Section 6.1:
6. Since the choice of routes from Boston to Detroit is independent of
the choice from Detroit to LA, the product rule says that we multiply
to get a total of 24 routes.
22. Suppose we want to find the numbers divisible by n tha
Solutions 6
Section 6.3
20: a) To have 3 zeros means to have 7 ones. To count you just have
to decide where the 0's go, for which there are 10 choose 3 ways.
b) There are 6 or 7 or 8 or 9 or 10 0's. So there are (10 choose
6) + (10 choose 7) + (10 choo
Solutions 7
6.5:
10: For the purpose of this solution, bagels are croissants. (They have
fewer letters to write.)
a) Choosing 12 bagels from 6 types is like writing down 12 "b"'s and
then separating them by 5 bars "": the bagels to the left of the
l
Solutions 9.
9.1.
16: I can't really draw this on the computer. The graph has one vertex
for each person and then one edge for each acquaintance.
20: Team 4 beat 3, but was beaten by everyone else.
9.2:
36: a) No. If the graph is simple, there can be no
Solutions 10

9.3:
40: Not isomorphic. The left graph has 4 vertices of degree 3, the
right one only 2.
42: Not isomorphic. If you select all vertices that have degree 4 in
both given graphs, and all edges that link such vertices, the lower
one give
Solutions 11.
9.5:
22: Since b has odd degree, no Euler circuit (directed or not) can
exist. If there is a directed Euler path, it must go from c to
b. Fleury's algorithm gives (for example)
c,b,c,e,b,f,a,f,d,e,f,e,a,b,d,c,b.
28: a) Whenever m and n
Solutions 12 to 9.7

5: no, it's version of K(3,3).
6: yes. push the afedge up. and the cdedge down.
7: yes, but hard to explain.
8: no. there are no triangles, so the shortest circuit has length 4.
but the inequality e < 2v3 is violated.
20: For ho
Math 373
Test 1
Spring 2012
February 16, 2012
1. Yue borrows 10,000 to buy a car. She will repay the loan with monthly payments for 6 years.
The annual effective interest rate on the loan is 15%.
Calculate Yues monthly payment.
2. Issac opens a new bank a