21-301 Combinatorics
Homework 1
Due: Wednesday, September 5
1. How many integral solutions of
x1 + x2 + x3 + x4 + x5 = 100
satisfy x1 6, x2 10, x3 3, x4 4 and x5 4?
2. Show that
n
n
k
k
=
n
2n .
k=0
3. How many ways are there of placing k 1s and 2n k 0s a
Exploring Combinatorics:
Inclusionexclusion
Like the textbook (section 3.7), we introduce inclusionexclusion with a silly problem.
Problem 1. A group of students attends three kinds of classes: art classes, biology
classes, and chemistry classes1 . It is
21-301 Combinatorics
Oleg Pikhurko
On Roots of Polynomials
Since we have used the following result in class twice, let me provide its proof.
Theorem. A non-zero polynomial P (x) = an xn + . . . + a1 x + a0 can have at most n roots
(counting their multipli
21-301 Combinatorics
Oleg Pikhurko
Exam 2
Exam 2 will take place during the class on October 18. Since we will start promptly at
9:30am, please arrive in time.
All exams are closed book and notes. Everything that we covered in the lectures (from the
begin
21-301 Combinatorics
Oleg Pikhurko
Root
L
L
R
L
Root
Root
R
R
L
R
L
R
Let un count the number of SPR trees on n vertices. For example, u1 = 1, u2 = 1, u3 = 2,
and u4 = 4. (Also, it is convenient to assume that u0 = 0.)
i) Show that for every n 1 we have
n
21-301: Exploring Combinatorics
Inclusionexclusion notes
Conclusion: Putting all of these together, we obtain that if angles in a spherical
triangle are , , and , then
area(T ) = area(S) area(T ) 3 1 area(S) + area(S)
2
+
360
or equivalently,
+ + 180
.
3
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2010: Test 4
Name:
Problem
1
2
3
Total
Points Score
40
40
20
100
1
Q1: (40pts)
How many ways are there of k-coloring the squares of the above picture if
the group acting is e0
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2010: Test 1
Name:
Problem
1
2
3
Total
Points Score
40
40
20
100
1
Q1: (40pts)
Prove that for 1 k n,
n
n
n
n
+
+ (1)k
k
0
1
2
= (1)k
n1
.
k
[Hint: Use induction on k.]
Soluti
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2012: Test 1
Name:
Andrew ID:
Problem
1 (a)
1 (b)
2
3
Total
Points Score
20
20
40
20
100
1
Q1: (40pts)
Let n, s be positive integers.
(a) Prove:
n
k=1
s1
k1
n
k
=
n+s1
.
n1
An
21-301 Combinatorics
Oleg Pikhurko
Problem 4 [2] How many integral solutions of
x1 + x2 + x3 + x4 + x5 = 36
satisfy x1 5, x2 0, x3 10, x4 10, and x5 13?
Problem 5 [1+1+3] Let m and n be positive integers.
i) Show that the number of functions f : [n] [m] s
21-301 Combinatorics
Oleg Pikhurko
Extra Reference Material on Generating Functions
[GKP] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete mathematics: a foundation
for computer science, Addison-Wesley Publ. Comp., 1989.
[N] I. Niven, Formal power se
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2005: Test 4
Name:
Problem
1
2
3
4
Total
Points Score
33
33
34
34
100
1
Q1: (33pts)
Consider the following general game involving one pile of chips . There is a
nite set of po
21-301 Combinatorics
Homework 2
Due: Friday, September 14
1. Show that for any n 0
0ikn
n
k
k
i
= 3n ,
where the sum goes over all integer pairs i, k such that 0 i k n.
2. In class we have dened the Catalan number Cn to be |PATHS (n, n)| and showed that
n
Homework 4
Due Wednesday, October 10th,
1. In the kingdom of Far Far Away there are coins of values 1, 2 and 3
dollars. In how many ways can the people of Far Far Away change n
dollars?
Hint:
1
=
1 + x + x2
1
3i
+
2
6
n=0
1
3i
2
2
n
+
1
3i
2
6
1
3i
+
2
2
21-301 Combinatorics
Homework 7
Due: Monday, November 12
1. Let rn = r(3, 3, . . . , 3) be the minimum integer such that if we n-color the edges of the
complete graph KN there is a monochromatic triangle.
(a) Show that rn n(rn
1
1) + 2.
(b) Using r2 = 6,
21-301 Combinatorics
Homework 8
Due: Monday, November 19
1. In a take-away game, the set S of the possible numbers of chips to remove is nite.
Show that the Grundy numbers g satisfy g (n) |S | where n is the number of chips
remaining.
2. Consider the foll
21-301 Combinatorics
Homework 9
Due: Monday, December 3
1. How many ways are there of k -coloring the squares of the above diagram if the group
acting is e0 , e1 , e2 , e3 where ej is rotation by 2 j/4. Assume that instead of 28 squares
there are 4n 4.
2.
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2005: Test 1
Name:
Problem
1
2
3
Total
Points Score
33
33
34
100
1
Q1: (33pts) Let a, m, n, p be positive integers. How many integer solutions
are there to
x1 + x2 + + x m = n
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2005: Test 2
Name:
Problem
1
2
3
Total
Points Score
33
33
34
100
1
Q1: (33pts) A box has four drawers; one contains three gold coins, one
contains two gold coins and a silver
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2005: Test 3
Name:
Problem
1
2
3
Total
Points Score
33
33
34
100
1
Q1: (33pts)
Let n, p be positive integers and let N = n2 p + 1. Suppose that x1 , x2 , . . . , xN
are real n
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2012: Test 2
Name:
Andrew ID:
Write your name and Andrew ID on every page.
Problem
1 (a)
1 (b)
2
3
Total
Points Score
20
20
40
20
100
Name:
Andrew ID:
Page 2
Q1: (40pts)
The s