Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 635 — #643
15.10. Combinatorial Proofs 635
Lemma 15.10.1 (Pascal’s Triangle Identity).
! ! !
n
D n 1 C n 1 : (15.10)
k k 1 k
We proved Pascal’s Triangle Identity without any algebra! Instead, we relied
purely on counting techniques.
15.10.2 Giving a Combinatorial Proof
A combinatorial proof is an argument that establishes an algebraic fact by relying
on counting principles. Many such proofs follow the same basic outline:
1. Define a set S.
2. Show that jSj D n by counting one way.
3. Show that jSj D m by counting another way.
4. Conclude that n D m.
In the preceding example, S was the set of all possible Olympic boxing teams. Bob
computed
! !
jSj D n 1 C n 1
k 1 k
by counting one way, and Ted computed
jSj D
n k
!
by counting another way. Equating these two expressions gave Pascal’s Identity.
Checking a Combinatorial Proof
Combinatorial proofs are based on counting the same thing in different ways. This
is fine when you’ve become practiced at different counting methods, but when in
doubt, you can fall back on bijections and sequence counting to check such proofs.
For example, let’s take a closer look at the combinatorial proof of Pascal’s Identity
(15.10). In this case, the set S of things to be counted is the collection of all
size-k subsets of integers in the interval Œ1::n.