2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/114014In this paper we count the number of paths and cycles in complete graphs by using the number $e$. Also, we compute the number of derangements in same way. Connection by $e$ yields some nice formulas for the number of derangements, such as $D_n=\lfloor\frac{n!+1}{e}\rfloor$ and $D_n=\lfloor(e+e^{-1})n!\rfloor-\lfloor en!\rfloor$, and using these relations allow us to compute some incomplete gamma functions and hypergeometric summations; these connections are hidden in the heart of a nice polynomial that we call it derangement function and a simple ordinary differential equation concerning it.12 pages, no figure, review of my works about the number of derangementsCombinatoricsNumber Theory40A25, 05C17, 05C38, 65L99, 33B20, 33C20, 26D15, 11J72, 20B40text