Introduction to Combinatorics and Its Applications
MATH 108

Winter 2015
Math 108 Combinatorics
Fall 2005
Homework 1 Solutions
(Problem 1: Derangements) Given n letters and n addressed envelopes, in how many
ways can the letters be placed in the envelopes so that no letter is in the correct envelope?
Solve this problem for the
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
MATH 108: Introduction to Combinatorics, Winter 2017
HOMEWORK 6
Due Tuesday, March 7
You should solve the homework on your own. Dont use any books or the internet.
Problem 1. Find the generating function of the sequence (1, 1, 0, 1, 1, 0, 1, 1, 0, . . .)
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
MATH 108: Introduction to Combinatorics, Winter 2017
HOMEWORK 1
Due Tuesday, January 24
You should solve the homework on your own. Dont use any books or the internet.
Problem 1. Find a construction of orthogonal
following Latin square:
0 1 2 3
1 0 3 2
2 3
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
MATH 108: Introduction to Combinatorics, Winter 2017
HOMEWORK 4
Due Tuesday, February 21
You should solve the homework on your own. Dont use any books or the internet.
Problem 1. Describe how to traverse all strings in cfw_0, 1n in such a way that each st
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
MATH 108: Introduction to Combinatorics, Winter 2017
HOMEWORK 5
Due Tuesday, February 28
You should solve the homework on your own. Dont use any books or the internet.
Problem 1. Find an example of a set system which satisfies the axioms (P 1), (P 2) of a
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
Math 108  Homework 3 Solutions
March 21, 2014
Problem 10C
Notice that
(n)
nx
x
=
n
(n)
nN
x
n
x
n
= 0 for n > x.
since (n)
x
We will prove that nN (n) n = 1 by induction on x. For x=1 we have that
(1) = 1. Now if the assumption is true for x 1, then for
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
Math 108  Homework 1 Solutions
April 7, 2014
Problem 1B
G is a simple graph with 10 vertices that is not connected. Thus, it has at least two connected
components. We have to show that:
Claim 1. The number of edges E(G) is no larger than 36.
Suppose th
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
Math 108  Homework 5 Solutions
May 14, 2014
Problem 13F
Proof 1: Do it by induction:
For n=2 it is easy to check. Assume that for Sn it is true that the number of permutations
with even number of cycles (denoted here as EV (n) is the same as the number o
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
Math 108  Homework 2 Solutions
April 16, 2014
Problem 2A
Recall that two graphs G and G are said to be isomorphic if there exists a bijection
: V (G) V (G )
such that
cfw_v, w E(G) cfw_(v), (w) E(G ),
that is: two vertices v, w of G are linked by an edg
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
Math 108  Homework 3 Solutions
April 23, 2014
Problem 1 On a planar graph, we have that 3f 2e since one face has at least 3 edges and
one edge has at most 2 faces. Therefore, using Eulers formula
1
1
deg(x) + v
2=f e+v e+v =
3
6 xV
which gives that id a.
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
Math 108  Homework 4 Solutions
May 7, 2014
Problem 1 The adjacency matrix of this graph is:
0 1 0
1 0 1
S=
0 1 0
1 0 1
1
0
1
0
and the characteristic polynomial of S is
p(x) = x2 (x2 4).
Therefore, the eigenvalues of the graph are 2, 0, 0, 2.
Problem
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
MATH 108: Introduction to Combinatorics, Winter 2017
HOMEWORK 2
Due Tuesday, January 31
You should solve the homework on your own. Dont use any books or the internet.
Problem 1. An orientation of an undirected graph G is a directed graph where we assign a
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
MATH 108: Introduction to Combinatorics, Winter 2017
HOMEWORK 3
Due Tuesday, February 7
You should solve the homework on your own. Dont use any books or the internet.
Problem 1. For a graph G = (V, E), we define its line graph L(G) where the vertex set is
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
MATH 108: Introduction to Combinatorics, Winter 2017
HOMEWORK 7
Due Tuesday, March 14
You should solve the homework on your own. Dont use any books or the internet.
Problem 1. Call two permutations 1 , 2 Sn conjugate (1 2 ) if there is a permutation
such
Introduction to Combinatorics and Its Applications
MATH 108

Winter 2015
Math 108 Combinatorics
Spring 2007
Homework 2 Solutions
1. Given n letters, of which m are identical and the rest are all distinct, nd a formula
for the number of words which can be made.
Solution. If we use i of the n m distinct letters and j of the m id
Introduction to Combinatorics and Its Applications
MATH 108

Winter 2015
Math 108 Combinatorics
Spring 2007
Homework 3 Solutions
1. How many positive integers less than 1000 have no factor strictly between 1 and 10?
Solution. A positive integer does not have a factor between 1 and 10 if and only if it
does not have a factor of
Introduction to Combinatorics and Its Applications
MATH 108

Winter 2015
Math 108 Combinatorics
Spring 2007
Homework 4 Solutions
1. Let (G, c) be a network with distinguished vertices s and t. A minimum cut in G is a
pair of sets (X, Y ) such that V = X Y , s X, t Y , X Y = , and c(X, Y ) is
minimal. Suppose (X, Y ) and (A, B)
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
Math 108 Problem Set 1 Solutions
February 1, 2017
Problem 1. We will solve the problem in the general n = 4m case. We copy the construction given in the problem for n = 8. Let the first two columns of our latin square
be
0
1
1
0
2
3
3
2
.
.
.
.
2k
2k + 1
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
Math 108 Problem Set 6 Solutions
March 7, 2017
P
n
Problem 1. The generating function of (1, 1, 1, 1, . . . ) is the geometric series
n=0 X =
1/(1X). To obtain the generating function that we want,
subtractP
the generating
Pwe must
3n+2
2
3 n
function of
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
Math 108 Problem Set 7 Solutions
March 8, 2017
Problem 1.
Symmetric: If 1 2 , so 2 = 1 1 for some permutation , then 1 = 2 1 =
( 1 )1 2 1 , so 2 1 .
Reflexive: If 1 = 2 , then we may take to be the identity permutation.
Transitive: Suppose that 1 2 and 2
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
Math 108 Problem Set 3 Solutions
February 5, 2017
Problem 1. Although you could ignore this problem, it turns out that the general problem
of determining whether L(G) has a Hamiltonian cycle is NPcomplete. This means that
the problem is conjecturally dif
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
Math 108 Problem Set 4 Solutions
February 19, 2017
Problem 1. We begin with a way of traversing all strings in cfw_0, 1n1 changing one bit at
a time. (This may be done by using the Gray code, for example.) Lets call the strings in
this list S1 , S2 , . .
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
Math 108 Problem Set 2 Solutions
February 1, 2017
Problem 1. The key point is to show that any such graph, if it contains any edges,
contains a cycle. Assuming this, we may complete the problem as follows. Let us proceed
by induction on the number of edge
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
Math 108 Problem Set 5 Solutions
February 26, 2017
Problem 1. Take a collection of 9 distinct collinear points Q1 , Q2 , . . . , Q9 in the plane.
Let l be the line through them. Choose a point R not on l. Then our set of points will be
Q1 , . . . , Q9 , R
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
HW # 5
DUE WEDNESDAY, MAY 14
vL followed by a number gives a problem from your text. Anything
marked as presentation, with a date, does not need to be handed in,
but you are encouraged to think about it before it is presented in class.
(0) Read Chapter 10
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
HW # 4
DUE WEDNESDAY, MAY 7
vL followed by a number gives a problem from your text. Anything
marked as presentation, with a date, does not need to be handed in,
but you are encouraged to think about it before it is presented in class.
(0) Read Chapter 13
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
HW # 3
DUE WEDNESDAY, APRIL 23
vL followed by a number gives a problem from your text. Anything
marked as presentation, with a date, does not need to be handed in,
but you are encouraged to think about it before it is presented in class.
(0) Read Chapter
Problem 1
If we remove two edges from K6 , there are two cases. First, if the two edges
have a vertex in common, then the subgraph on the other ve vertices is K5 .
This is clear because we removed no edges between any of these ve vertices.
Otherwise, if t
Homework 8  Math 108, Frank Thorne (fthorne [at] math.stanford.edu)
Due Friday, December 3, at 4:30 p.m.
Please note: This homework looks long, but a lot of the questions are for extra credit.
Why give multiple proofs of the same result when, clearly, on