CS173: Discrete Mathematical Structures
Fall 2005
Homework exam problem
Due 11/27/05, 8a
Give a combinatorial proof that
!
2n
n
"
#
$
%
&
' =
n
k
"
#
$
%
&
'
k=0
n
(
2
.
a. Briefly describe the collections of items counted by the expression on the left side
of the equation.
b. Prove that
!
n
k
"
#
$
%
&
' =
n
n(k
"
#
$
%
&
'
.
c. Use part b. to explain the expression
!
n
k
"
#
$
%
&
'
2
for a particular k.
d. Explain the role of the summation on the right side of the expression.
e. Complete the proof by connecting your explanations from part a. and part d.