Combinatorics Through Guided Discovery, 2004a

4.1. The Idea of Generating Functions 79
that the coefficient of x k in
(
∑ ∞
)
a i x i
∞∑
b j x j
i=0
j=0
is the number of ordered pairs (x 1 , x 2 ) in S 1 × S 2 with total value k, that is
with v 1 (x 1 )+v 2 (x 2 )=k. This is called the product principle for generating
functions. (h)
Problem 191 may be extended by mathematical induction to prove our next
theorem.
Theorem 4.1.1. If S 1 , S 2 ,...,S n are sets with a value function v i from S i to the nonnegative
integers for each i and f i (x) is the generating function for the number of elements of
S i of each possible value, then the generating function for the number of n-tuples of each
possible value is ∏ n
i=1 f i(x).
4.1.6 The extended binomial theorem and multisets
• Problem 192. Suppose once again that i is an integer between 1 and n.
(a) What is the generating function in which the coefficient of x k is 1? This
series is an example of what is called an infinite geometric series. In
the next part of this problem, it will be useful to interpret the coefficient
one as the number of multisets of size k chosen from the singleton set
{i}. Namely, there is only one way to choose a multiset of size k from
{i}: choose i exactly k times.
(b) Express the generating function in which the coefficient of x k is the
number of multisets chosen from [n] as a power of a power series.
What does Problem 125 (in which your answer could be expressed
as a binomial coefficient) tell you about what this generating function
equals? (h)
◦ Problem 193. What is the product (1 − x) ∑ n
k=0 xk ? What is the product
(1 − x)
∞∑
x k ?
k=0
Problem 194. Express the generating function for the number of multisets
of size k chosen from [n] (where n is fixed but k can be any nonnegative
integer) asa1oversomething relatively simple.