As we have seen, a typical counting problem includes one or more parameters, which of course show up in the solutions, such as $n\choose k$, $P(n,k)$, or the number of derangements of $[n]$. Also recall that $$(x+1)^n=\sum_{k=0}^n {n\choose k}x^k.$$ This provides the values ${n\choose k}$ as coefficients of the Maclaurin expansion of a function. This turns out to be a useful idea.
Definition 3.0.1 $f(x)$ is a generating function for the sequence $\ds a_0,a_1,a_2,\ldots$ if $$f(x)=\sum_{i=0}^\infty a_i x^i.$$ $\square$
Sometimes a generating function can be used to find a formula for its coefficients, but if not, it gives a way to generate them. Generating functions can also be useful in proving facts about the coefficients.