MATH 431-200/500. Structures and Methods in
Combinatorics
Extra-credit Assignment 3.
Due on Wednesday, May 7th, 2008
Please show your argument and computation. Calculators and comput-
ers are not permitted.
1. Show that a simple graph (no loops or multiple edges) with nvertices
is connected if it has more than (n−1)(n−2)/2 edges.
2. Give a combinatorial proof that
n
X
k=1
kn
k2
=n2n−1
n−1.
Hints: Count in two ways the number of ways to select a committee,
with nmembers from nmathematicians and ncomputer scientists,
such that the chairperson is a mathematician.
3. How many binary strings of length n, where n≥4, contains exactly
two occurrences of 01?
4. Consider an n-by-nboard in which there is a nonnegative number aij
in the square in row iand column j, (1 ≤i, j ≤n). Assume that
the sum of the numbers in each row and in each column equals 1.
Prove that it is possible to place nnon-attacking rooks on the board
at positions occupied by positive numbers.
5. An integer is called parity-monotonic if its decimal representation
a1a2. . . aksatisfies ai< ai+1 if aiis odd, and ai> ai+1 if aiis even.
How many four-digit parity-monotonic integers are there?