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 X has 2 n + 1 elements, find the number of subsets of X with at most n elements.
Problem # 7
How many words of length 4 can be made from the letters of the word MIS-
SISSIPPI ?
Problem # 8
Prove that f
1
2
+ f
2
2
+ · · · + f
2 n
= f n f n +1 whenever n is a positive integer.
Problem # 9
Prove that f
1
+ f
3
+ · · · + f
2 n − 1
= f
2 n whenever n is a positive integer.
Problem # 10
Show that f n
+1 f n − 1
− f
2 n
= ( − 1) n whenever n is 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 , . . .