Math 484
Problem Set #5 Solution
7 March 2014
1. Let S10 denote the set of permutations of 10.
(a) How many 5-cycles are there in S10 ?
Solution: Here is a solution in the spirit of Bender & Williamson. Let N be
the number of 5-cycles. We count the number
Math 484
Problem Set #1 Solution
5 February 2014
1. Use the Rule of Product (Theorem 1.2) to count the number of even four-digit positive
integers. Explain how the even four-digit positive integers correspond to a particular
Cartesian product.
Solution: W
Math 484
Problem Set #3 Solution
24 February 2014
1. (a) Use a spreadsheet program to construct a table of the Stirling numbers of the
cfw_
n
second kind,
, for 1 n 10, and 1 k 10.
k
(This was the last exercise on our computer lab worksheet.)
Solution: He
Math 484
Problem Set #6 Solution
14 March 2014
1. We decorate the vertices of a regular pentagon using one white bead, two black beads,
(
)
5
and two red beads. If the pentagon is xed, then there are
= 30 ways to do
221
this.
Let S denote the set of these
Math 484
Problem Set #2 Solution
12 February 2014
1. Consider an ordinary deck of 52 playing cards supplemented by two wild cards, W1 and
W2. In a poker hand, a wild card can be used as any card (including those in a fth
suit), but it must always be used
Math 484
Problem Set #4 Solution
28 February 2014
1. (B&W Exercise 1.4.8) For each n > 0, let Bn denote the total number of partitions of
n. That is,
cfw_ cfw_
cfw_
n
n
n
Bn =
+
+ +
.
1
2
n
For convenience, we dene B0 = 1. There is one partition of cfw_,
Math 484
Problem Set #7 Solution
28 March 2014
1. A cube has twelve edges. The symmetry group of the cube has
24 elements. One element is the identity, and there are four other
types of elements (technically, these are four conjugacy classes).
(a) For eac