Remaining Gum

Omar has $N$ pieces of 5gum in his left pocket and $N$ pieces of Orbit gum in his right pocket. He's dealing out gum to $M > 2N$ people in his probability class. Every time he deals out a piece of gum to someone, he picks which pocket to deal from uniformly at random. When Omar is asked to give gum and he looks in his pocket to find there are no pieces of gum remaining in it, he counts up the number of pieces of gum in his other pocket. Find the probability there are exactly $k$ pieces of gum in the other pocket, $0 \leq k \leq N$, when $N = 8$ and $k = 4$.