We call a cycle of length $k$ with values in $\llbracket 1,n \rrbracket$, any $(k+1)$-tuple $\vec{\imath} = (i_{1}, i_{2}, \ldots, i_{k}, i_{1})$ of elements of $\llbracket 1,n \rrbracket$. We denote $|\vec{\imath}|$ the number of distinct vertices of the cycle $\vec{\imath}$.
Show that the number of cycles of length $k$ in $\llbracket 1,n \rrbracket$ passing through $\ell$ distinct vertices is at most $n^{\ell} \ell^{k}$.