Combinatorial problem with $12$ unique cards The problem is "how many ways can we distribute $12$ unique cards to $4$ persons with every person receiving at least $1$ card" Without the requirement that everyone getting at least $1$, it would be $4^{12}$, and my initial thought was that with the req. It would be $_nP_r(12,4)\times 4^8$ but that's way too high, looks like I need to divide with some permutations that's already accounted for.. Please help!

Each admissible allocation is a surjective map $f:\>[12]\to[4]$. Such maps are counted by the Stirling numbers of the second kind. To be precise: The number of surjective maps $f:\>[n]\to[k]$ is given by $k!\, S(n,k)$. There are recursions and tables for these numbers, but no closed formula in terms of factorials etc.