Dr. A. Betten Fall 2009
MATH 501 Introduction to Combinatorial Theory
Assignment # 2
Problem # 5
A box contains 20 cell phones, of which 4 are Nokia, 7 are Motorola and 9 are
Samsung. What is the smallest number of cell phones which must be chosen
(blindfolded) so that the selection is guaranteed to contain r= 4,5,6,7,8,9
phones of the same make?
Problem # 6
If a set Xhas 2n+ 1 elements, find the number of subsets of Xwith at most
nelements.
Problem # 7
How many words of length 4 can be made from the letters of the word MIS-
SISSIPPI ?
Problem # 8
Prove that f2
1+f2
2+· · · +f2
n=fnfn+1 whenever nis a positive integer.
Problem # 9
Prove that f1+f3+· · · +f2n−1=f2nwhenever nis a positive integer.
Problem # 10
Show that fn+1fn−1−f2
n= (−1)nwhenever nis a positive integer.
Problem # 11
Write a closed-form generating function for each of the following sequences:
a) 1,−1,1,−1,1,−1, . . .
b) 1,0,1,0,1,0, . . .
c) 1,1,1,1,1,1, . . .
d) 1,1,−1,−1,1,1,−1,−1, . . .
e) 1,0,−1,0,1,0,−1,0, . . .
f) 1,0,0,1,0,0,1,0,0, . . .
