Combinatorics is, relatively speaking, a new field of mathematics. And as with any new field of study, it still remains very unorganized. How unorganized? There are many many topics, subtopics, side-topics, stand-alone problems and ungeneralized results laying everywhere, just like my room. One can witness this very early in combinatorics. Many of the math competition problems are about counting, and every problem seems to require a clever trickery in order to solve. For this reason many mathematicians tend to frown on combinatorics as simply some random collection of brainteasers, and not a "deep" field to study. Nonetheless, I think combinatorics is a promising field. Timothy Gowers, Fields medalist, is an example of a supporter of combinatorics. In fact, his award-winning research made surprising and insightful links between problems in combinatorics and other fields of mathematics.
The topic of generating functions of sets is an example of combinatorics boiling down many counting problems. Before talking about generating functions, let's look at a motivation from computer science. There are many more motivations, but this one I think fits best my knowledge at the moment.
In computer science, one pillar of research is dedicated to formal languages (automata theory). It is clearly an important topic since it studies the representation of information and of machines that computes with it as input. This field answers questions such as computability, and gives a solid theoretical framework for actually creating good computer languages along with their compilers and parsers.
Among the many classes of languages, one is called a "regular language".
For example, $\Sigma = \{0, 1\}$ is an alphabet. $101101$ and $\epsilon$ are words over that alphabet, and $L = \{101101, \epsilon\}$ can be a language. We allow languages with infinitely many words such as $L = \{\epsilon, 0, 1, 00, 01, 10, 11, 100, 101, \cdots\}$, which is basically every single possible word from the alphabet. Notice if we took out the words with leadings zeros except the first zero, we get the language of binary numbers.
Definition: A regular language $L$ is a set of words $w$ which are finite strings of symbols over a finite alphabet $\Sigma$. A word can be the empty word, which we denote $\epsilon$. A language can contain uniquely the empty word $L = \{\epsilon\}$, or can contain nothing, the empty language $L = \emptyset$.
A language $L$ can be formed from other languages say $L_a$ and $L_b$ from three rules.
Union: $L = L_a \cup L_b$, where $L = \{a | a \in L_a$or$a \in L_b\}$.
Concatenation: $L = L_aL_b$, where $L = \{ab | a \in L_a, b \in L_b\}$
Repetition: $L = \{L_a\}^{\star}$, where $L = \{\epsilon \cup L_a \cup L_aL_a \cup \cdots\}$
The base cases are sets containing one element in the alphabet.
For example, $\{0, 1\}^{\star}$ is the set of all binary strings (0's and 1's, whereas $\{0, 11\}^{\star}$ is the set of all binary strings with even length blocks of 1's.
