Math 353 Fall 2014 - Homework #2 Solutions
2.5.2B There are 3 Ps, 3Os, two E, N, Rs, and one I, M, S. So we want the multnomial coecient:
15
3, 3, 2, 2, 2, 1, 1, 1
And yes properispomenon is a real word!
=
15!
= 4540536000.
3!3!2!2!2!
2.6.2B For cycle typ
Fall 2014: Math 353- Introduction to Combinatorics I,
Course Information Sheet
Instructor:
David Hemmer
Office:
211 or 226 (chairs office) Mathematics Building
Email:
dhemmer@math.buffalo.edu
Office Phone:
645-8775
Class Meetings:
MWF 9:00-9:50 a.m.
Offic
Math 353 Homework #9- SOLUTIONS
1. 11.5.5B. The order of an element is the lcm of the cycle lengths. So we need a partition =
(1 , 2 , . . . , t ) so 1 + 2 + + t = 50 with the lcm of the i as large as possible. Some trial and
error will show the biggest i
Math 353 Fall 2014 - Homework #7 Solutions
6.4.3BLet 2n + k have exactly n + k parts. Then the rst column in the Ferrers diagram of has exactly n + k dots,
and removing the rst column leaves the diagram of a partition with 2n + k (n + k) = n dots. Similar
Math 353 Homework #6- Due Wednesday 10/8/14-SOLUTIONS
1. Proof: Let m be the number of partitions of n into exactly k parts (m = pk (n) = pk1 (n).
Suppose we have a solution of x1 + x2 + + xk = n with all the xk positive. We know
there are C(n 1, k 1) sol
Math 353 Fall 2014 - Homework #1 Solutions
2.2.4B There are 8! ways to put the red counters on the board. For each, we need to determine how many ways to put the
green counters on so we have a legal arrangement. Given any such arrangement there is a uniqu
Math 353 Homework #5- Due Wednesday 10/1/14- SOLUTIONS
1. Lets dene X(n, k) to be all the possible products of n k integers taken from cfw_1, 2, . . . , k, repeats allowed.
We must show that for 1 k < n that
a = S(n, k).
aX(n,k)
We proceed by induction on
Math 353 Exam Review Sheet
The exam will cover from Chapter 2-7.5 plus other things done in class.
Definitions: You should know how these quantities are defined and be able to calculate them in small cases:
P(n,r)=n!/(n-r)!- the number of ways to make an
Math 353 Homework #10- Due Monday 11/24/14- SOLUTIONS
1. 13.2.1B The regular octagon has 16 symmetries. Let r be the clockwise rotation by /4, so there are
8 rotations, namely cfw_e, r2 , r3 , . . . , r7 . There are 8 reections, 4 of them have axes connec
gowf/M/f
Math 353 Midterm Exam - October 17, 2014
Instructions: You may not use any notes, books, calculators, etc. It is ok if your nal answers
include binomial coefficients and if you do not multiply out exponentials.
1. (55 points) Short answer, little
Math 353 Fall 2014 - Homework #4 Solutions
3.3.8B We are looking at 6 digit integers from 000000 to 999999. The key observation is that once we know the six digits,
there is exactly one corresponding integer, just arrange the digits in nondecreasing order
Name:
Math 353 Quiz #9 - November 21, 2014
1. Let G act on a set X. Give the Frobenius (Burnside) formula for the number of orbits.
number of orbits =
1
|F ix(g)|.
|G| gG
2. Dene what it means for a group G to act on a set X.
G acts on X if there is a map
Name:
Math 353 Quiz #8 - November 10, 2014
1. Dene the center of a group G.
Z(G) = cfw_z G | zg = gzg G.
2. Let g be an element of a group G. The conjugacy class of g is . . . .?
The set of elements xgx1 as x runs over all elements of G.