Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 681 — #689
16.2. Counting with Generating Functions 681
(see Problem 16.5), so
Œx n 1 d
n
1
D
.1 x/ k d n .0/ 1 x .1 x/ k nŠ
k.k C 1/ .k C n 1/.1 0/
.kCn/
D
!
nŠ
n C .k 1/
D :
n
In other words, instead of using the donut-counting formula (16.10) to find the
coefficients of x n , we could have used this algebraic argument and the Convolution
Rule to derive the donut-counting formula.
16.2.5 The Binomial Theorem from the Convolution Rule
The Convolution Rule also provides a new perspective on the Binomial Theorem
15.6.4. Here’s how: first, work with the single-element set fa 1 g. The generating
function for the number of ways to select n different elements from this set
is simply 1 C x: we have 1 way to select zero elements, 1 way to select the one
element, and 0 ways to select more than one element. Similarly, the number of
ways to select n elements from any single-element set fa i g has the same generating
function 1Cx. Now by the Convolution Rule, the generating function for choosing
a subset of n elements from the set fa 1 ; a 2 ; : : : ; a m g is the product .1 C x/ m of the
generating functions for selecting from each of the m one-element sets. Since we
know that the number of ways to select n elements from a set of size m is m n
, we
conclude that that
!
Œx n .1 C x/ m D
m ;
n
which is a restatement of the Binomial Theorem 15.6.4. Thus, we have proved
the Binomial Theorem without having to analyze the expansion of the expression
.1 C x/ m into a sum of products.
These examples of counting donuts and deriving the binomial coefficients illustrate
where generating functions get their power:
Generating functions can allow counting problems to be solved by algebraic
manipulation, and conversely, they can allow algebraic identities to be derived by
counting techniques.