Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 633 — #641
15.10. Combinatorial Proofs 633
for any nonempty set I Œ1::m. This lets us calculate:
m[
jSj D
C
ˇ
pi
ˇˇˇˇˇ
so
D
D
D
D
iD1
X
;¤I Œ1::m
X
;¤I Œ1::m
n
n
X
;¤I Œ1::m
mY
iD1
\
. 1/
ˇˇˇˇˇ
jI jC1 C pi
ˇˇˇˇˇ
i2I
. 1/ jI jC1 n
Q
1
j 2I p j
1
Q
j 2I . p j /
1
p i
! C n;
.n/ D n
jSj D n
mY
iD1
(by (15.8))
(by Inclusion-Exclusion (15.6))
1
1
;
p i
(by (15.9))
which proves (15.7).
Yikes! That was pretty hairy. Are you getting tired of all that nasty algebra? If
so, then good news is on the way. In the next section, we will show you how to
prove some heavy-duty formulas without using any algebra at all. Just a few words
and you are done. No kidding.
15.10 Combinatorial Proofs
Suppose you have n different T-shirts, but only want to keep k. You could equally
well select the k shirts you want to keep or select the complementary set of n k
shirts you want to throw out. Thus, the number of ways to select k shirts from
among n must be equal to the number of ways to select n k shirts from among n.
Therefore:
!
n n
D
k n k
!
:
This is easy to prove algebraically, since both sides are equal to:
nŠ
kŠ .n k/Š :