MATH 5705: HOMEWORK 9 SOLUTIONS
Chapter 8
Problem 1. Let hn be the number of pairings as described in the problem. Give
the points labels from 1 to 2n going around the circle. First, note that it is not
important that the points are evenly spaced, or even
MATH 5705: HOMEWORK 5 SOLUTIONS
Chapter 6
Problem 1. Let A, B , and C be the set of integers in the given range that are
divisible by 4, 5, and 6 respectively. We have seen the number of integers from 1 to
n divisible by m is n/m . Therefore
|A| = 10000/4
MATH 5705: HOMEWORK 4 SOLUTIONS
Chapter 4
Problem 44. Let x X be given. Then x Ai for some i. Since x Ai and
x Ai we have xRx. So R is reexive.
Let x, y X and suppose xRy . Then there exists an i such that x Ai and
y Ai . As y, x Ai we know yRx. So R is s
MATH 5705: HOMEWORK 3 SOLUTIONS
Chapter 4
Problem 5. The number k = b1 + b2 + . . . + bn gives the total number of inversions
in the permutation, i.e., the total number of pairs of integers for which the larger
appears rst. Switching two adjacent numbers
MATH 5705: HOMEWORK 2 SOLUTIONS
Chapter 2
Problem 60. I assume we are choosing an order (e.g. 3 bagels of type 1, 0
bagels of type 2, . . . ) at random from the set of legal orders, rather than choosing
each bagel ranomdly one at a time. A legal order con
MATH 5705: HOMEWORK 1 SOLUTIONS
Chapter 6
Problem 24. (a) Let rk be the number of ways to put k non-attacking rooks on
the Xed out squares. Consider the six Xs to be in three zones of two each, according
to row. Then two rooks placed on Xs attack each oth
MATH 5705: MIDTERM EXAM
Instructions
Now would be a good time to write your name at the bottom of this page.
Once instructed to start, you will have 90 minutes for this exam.
You may not use your textbook, notes, or any other resource. If evaluating
an
MATH 5705: HOMEWORK 11 SOLUTIONS
Chapter 14
Problem 14. There are 23 = 8 total 2-colorings which can be groups by equivalence as follows:
R
RR
R
R
W
RW
WR
RR
R
W
W
WW
RW
WR
W
WW
Hence there are 4 nonequivalent colorings. Given 3 colors there are 33 = 27 t
MATH 5705: HOMEWORK 10 SOLUTIONS
Chapter 8.
Problem 13. For concreteness, let X = cfw_1, 2, . . . , p and Y = cfw_1, 2, . . . , k .
Recall that
k !S (p, k )
equals the number of ordered partitions of X into k non-empty subsets A1 , A2 , . . . , Ak .
Given
MATH 5705: HOMEWORK 8 SOLUTIONS
Chapter 7
Problem 26. Let hn be the number of colorings as described in the question, and
let
xn
g (x) =
hn
n!
n 0
be its exponential generating function. We have seen that g (n) factors into four
pieces, each a sum of xj /
MATH 5705: HOMEWORK 1 SOLUTIONS
Chapter 2
Problem 6. Consider counting all k -digit numbers with properties (a) and (b)
by selecting the digits one by one, starting on the left. The leftmost digit can be
anything other than 0 (otherwise it wouldnt be a k-