Dr. A. Betten
Fall 2005
M501 Combinatorics I
exercise sheet # 2
Exercise # 1
Prove the following identities:
n k
n n−l
a)
=
,
k
l
l
k−l
(3 points)
k X
m
n
m+n
b)
=
(recall that
i
k−i
k
i=0
c)
k X
n+i
i
i=0
n
X
=
n
k
= 0 for k < 0 or k > n),
n+k+1
,
k
2 n
0
(−1)
d)
=
m
(−1)
k
k=0
k
if n is odd
if n = 2m.
2m
m
Exercise # 2
Let Fn be the n-th Fibonacci number. Prove the following identities:
(3 points)
a) Fn2 − Fn+1 Fn−1 = (−1)n for n ≥ 1.
b)
n
X
Fi = Fn+2 − 1.
i=0
2
+ Fn2 = F2n ,
c) Fn−1
bn/2c d) Fn =
X
i=0
Fn−1 Fn + Fn Fn+1 = F2n+1 .
n−i
.
i
Exercise # 3
(1 points)
Let Fn be the n-th Fibonacci number. Show that Fn is composite for all odd n > 3.
Exercise # 4
(3 points)
If n is a multiple of 8, show that the number of subsets
of an n-set of size divisible by 4 is
2n−2 + 2(n−2)/2 . Example: for n = 8, we have 80 + 84 + 88 = 1 + 70 + 1 = 72 = 64 + 8 = 26 + 23 .
√
Hint: consider (1 + i)n , where i = −1.
due to Friday, 9/23/05.