  In statistical mechanics, the generally called Stirling approximation is
actually an approximation of Stirling's formula. In this article, it is shown
that the term that is dropped is in fact the one that takes fluctuations into
account. The use of the Stirling's exact formula forces us to reintroduce them
into the already proposed solutions of well-know puzzles such as the
extensivity paradox or the Gibbs' paradox of joining two volumes of identical
gas. This amendment clearly results in a gain in consistency and rigor of these
solutions.
