MATH 481 MIDTERM EXAM SOLUTIONS
(1) Give two proofs (algebraic and combinatorial) of the following formula:
n+1
k+1
k+1
1
=
n+1
1
n
k
Proof. Algebraic:
n+1
k+1
k+1
1
n+1
1
= (k + 1)
n
k
(n + 1)!
(n + 1)!
=
(k + 1)!(n + 1 (k + 1)!
k !(n k )!
= (n + 1)
n!
(
MATH 310: EXTRA CREDIT PROBLEMS
(1) (10 points, due 8/12) For which primes p is the polynomial x2 + 1 irreducible
in Zp [x]? Equivalently, when does the equation x2 = 1 have no solutions mod
p? You do not have to prove anything. Just do some examples and
MATH 310: HOMEWORK 1
(1) 1.1 # 6.
(2) 1.1 # 7.
(3) 1.2 # 10.
(4) 1.2 # 32.
(5) Prove that if n Z, then n(n + 1)(n + 2) is divisible by 6.
(6) 2.1 # 12.
(7) 2.2 # 2.
(8) 2.3 # 7 (b),(d). Use the reverse Euclidean algorithm.
(9) List all the positive diviso
MATH 481: HOMEWORK 1
(1) (a) How many possible outcomes are there if we roll a six-sided die two times
or ip a coin three times?
(b) How many ways are there to ip a coin ten times and get heads at least six
times?
(2) Sec. 2.1 #2.
(3) Given a real number
MATH 481: HOMEWORK 2
(1) Let a, b, and c be positive integers. How many paths are there from (0, 0, 0) to
(a, b, c) if we are only allowed to increase one of the coordinates by one at each
step?
(2) Let n be a positive integer. Give two proofs of the foll
MATH 481: HOMEWORK 7
(1) Let G be a non-empty graph of order n whose vertices have degrees d1 , . . . , dn .
The line graph of G is dened as follows: the vertices of L(G) are the edges of
G, and two vertices of L(G) are adjacent if they share an endpoint
MATH 481: HOMEWORK 8
(1) A saturated hydrocarbon is a connected graph where each vertex has degree 1
(hydrogen atom) or degree 4 (carbon atom) and the number of hydrogen
atoms is two more than twice the number of carbon atoms. Prove the following
statemen
MATH 481: HOMEWORK 9
(1) Show that C6 is a polyhedron and draw its dual polyhedron.
(2) Suppose a polyhedron is composed of 120 triangles and 160 squares. How many
vertices and edges does it have?
(3) Find the chromatic number of L(Kn,n ).
Hint: Try an ex
MATH 481: HOMEWORK 6
(1) Use the rst theorem of graph theory to nd the order and size of the complete
bipartite graph Kn,m .
(2) (30 points) The friendship graph is dened as follows: the vertices are people,
and we draw an edge between two people if they
MATH 481: HOMEWORK 5
(1) Give two proofs (algebraic and combinatorial) of the following formula:
n
n
+
2
1
m
m
+
1
2
=
n+m
2
(2) Consider a 4 4 chessboard (i.e. a square divided into 4 rows and 4 columns)
with 4 chess pieces on it. There are 5 possible co
MATH 481: HOMEWORK 3
(1) Give two proofs (algebraic and combinatorial) of the following formula:
n
2
2
=3
n
n
+3
3
4
(2) Let n be a positive integer. Prove the following formula for any real number
x = 1:
n1
1 xn
xk =
1x
k=0
(3) Find (and prove) a formula
MATH 481: HOMEWORK 4
(1) Suppose a0 = 0, a1 = 8, and ak = 2ak1 + 3ak2 for k 2. Write down the
generating function G(x) as a quotient of two polynomials and nd an explicit
formula for ak .
(2) (a) Express the following function as the product of a polynomi
MATH 310: HOMEWORK 2
(1) 1.1 # 8.
(2) 1.2 # 24.
(3) 1.3 # 23.
(4) 1.3 # 28
(5) 2.1 # 16.
(6) 2.2 # 8.
(7) 2.2 # 11.
(8) Use the Chinese remainder theorem to nd the 4-digit number x such that:
x 4 mod 9
x 5 mod 11
x 27 mod 101
(9) Find all positive integer
MATH 310: HOMEWORK 3
(1) Determine if the following statement is true or false: If n is a positive integer
not divisible by 41, then n2 + n + 41 is prime. Justify your answer.
(2) 3.1 #9.
(3) 3.1 #10.
(4) 3.1 # 13 (b).
(5) 3.1 # 24. Compare your answer to
MATH 310: HOMEWORK 6
(1) Solve 13x = 11 in Z83 . Can you solve 99x = 1 in Z110000 ?
(2) Let m, n Z such that gcd(m, n) = 1. Use Theorem 1.5 to prove that
(m) (n) = (mn)
Is this formula still true if gcd(m, n) = 1?
(3) Use the Chinese Remainder Theorem to
MATH 481: HOMEWORK 1 SOLUTIONS
(1) There are 20 runners in a qualifying race for the U.S. olympic team. The team
will consist of the top ve nishers. Answer the following questions by deciding
which type of counting problem is involved.
(a) How many possib
MATH 481: HOMEWORK 2 SOLUTIONS
(1) Let n be a positive integer. Give an algebraic proof and a combinatorial proof
of the following:
n
n
+
3
2
2
n
+
1
1
=
n+2
3
Algebraic proof:
n
n
+
2
3
=
n
2
+
1
1
=
n(n 1)(n 2) 2n(n 1)
+
+n
6
2
n(n2 3n + 2 + 6n 6 + 6)
n
MATH 481: HOMEWORK 3 SOLUTIONS
(1) Give two proofs (algebraic and combinatorial) of the following formula:
n
2
2
n
3
=
3
1n
+
1
24
4
2
Algebraic proof:
n
2
2
=
=
n
2
(
n
2
2
1)
=
n(n 1)
2
1
2
n(n 1)
1
2
n(n 1)(n + 1)(n 2)
n+1
n
n
n(n 1)(n2 n 2)
=
=3
=3
+
MATH 481: HOMEWORK 4 SOLUTIONS
(1) If our currency consists of a two-cent coin and three kinds of pennies, how many
ways can we make change for a dollar?
Answer: We are looking for the number of ways to put 100 identical objects
into 4 bins, where one of
MATH 481: HOMEWORK 5 SOLUTIONS
(1) Consider a 5 5 chessboard (i.e. a square divided into 5 rows and 5 columns)
with 5 pawns on it. There are 7 possible congurations:
(a) There is a row with 5 pawns.
(b) There is a row with 4 pawns, and another row with 1
MATH 481: HOMEWORK 6 SOLUTIONS
(1) Let G be a graph of order n and size k . Assume that the vertices are labeled
1, . . . , n and the edges are labeled 1, . . . , k . The incidence matrix of G is the
n k matrix which has a 1 in the (i, j ) entry if the i-
MATH 481: HOMEWORK 7 SOLUTIONS
(1) For any graph G = (V, E ), dene the complement graph G = (V, E ) so that
/
cfw_a, b E if and only if cfw_a, b E . Prove that s(G) = s(G).
Hint: Show that every symmetry of G is a symmetry of G, and vice versa.
Proof. Sin
MATH 481: HOMEWORK 8 SOLUTIONS
(1) (20 points)
(a) Find all the isomers of hexane i.e. nd all saturated hydrocarbons with
molecular formula C6 H14 .
(b) Find the number of symmetries of each molecule.
(c) Find the number of labelings (up to symmetry) of e
MATH 481: HOMEWORK 9 SOLUTIONS
(1) Let G = (V, E ) be a graph. Dene the line graph L(G) as follows:
L(G) = (E, W ) where cfw_a, b, cfw_c, d W if and only if |cfw_a, b cfw_c, d| = 1.
Prove that W is a disjoint union of edge sets of maximal complete subgrap
MATH 310: HOMEWORK 4
(1) Let R be ring with identity. Prove that there exists exactly one ring homomorphism f : Z R such that f (1) = 1.
(2) Let R be an integral domain of characteristic n > 0. Prove that n is prime.
(3) Let R be a ring with identity. Pro
MATH 310: HOMEWORK 5
(1) Let R be a ring. Use the First Isomorphism Theorem to prove that R/(0) R
=
and R/R 0. Use these facts to describe the rings Z1 and Z0 .
=
(2) Let F be a eld and let f (x) F [x], f (x) = 0. Let I = (f (x) be the ideal
generated by
MATH 481: HOMEWORK 10
(1) Determine whether each of the following statements is true or false. Justify
your answer.
(a) Every bipartite graph is planar.
(b) Every planar graph is bipartite.
(c) A tree whose size is even must have a vertex of even degree.