The problem is:
How many are ways to deal a deck of 52 cards to 4 players, and every player has at least one card?
The answer with inclusion-exclusion principle is: $4^{52}-4\cdot 3^{52}+6\cdot 2^{52}-4 $
But I'm wondering, why isn't it equal to the number of solutions: $a+b+c+d=52, \ a,b,c,d\in\mathbb{N}, a,b,c,d>0$ which we can calculate by finding coefficient before $x^{52}$ in expansion to series this function: $\frac{x^4}{(1-x)^4}=\left(\sum_{n\ge 1}x^n\right)^4$ which is quite easy? The result is completely different(extremely less). What we don't count this way, what we are missing? Can this approach be fixed?