CS173 Discrete Mathematical Structures
Fall 2006
Homework #9
due Tuesday, November 7, 2006, 8:00 a.m.
(46 points + 6 bonus points)
1. The Pigeonhole Principle (10 points)
a. (5 points) There are 26 houses on a street. Each house has a distinct address between
1000 and 1049, inclusive. Show that at least two houses have addresses that are
consecutive integers.
Hint: Group all available addresses into pairs of consecutive numbers. Take them as
“pigeonholes”.
b. (5 points) Prove that any set of 17 nonempty strings over { a, b, c, d} have two different
strings whose starting letters agree and ending letters agree.
Hint: Nonempty strings over {a, b, c, d} can be any size greater than 0, can use any
combination of letters, and can have any order. We are only concerned with the starting
and ending letters of strings.
2. Combinatorial Proof (8 points)
Give a combinatorial proof that if n is a positive integer then
2 2
0
( 1)2
nn
k
n
k n n
k
.
Hints:
1. Read Chapter 5.4 in your textbook. There are a couple of nice combinatorial proofs.
See Corollary 1 and Theorem 2.
2. Consider a set A of n elements. Both sides count the ways to first select a subset S of
A and then select two not necessarily distinct elements from S. By saying not
necessarily distinct, it means that we select twice from S, and the selected element
of each selection may be the same element. For example, we choose x for the first
time, and again choose x for the second time.
3. Express the right-hand side as
2 1
( 1)2 2
n n
n n n
. Match these two terms with the
following two situations: the two elements chosen are different, and the two
elements chosen are the same.
