Balls and Boxes

You are given $N+1$ boxes. The $i_{th}$ box contains $A_i$ balls. Balls from the same box are identical, while balls from different boxes are distinct. You need to arrange all the balls in a row such that before the occurrence of any ball from the $N+1$-th box, at least one ball from each of the first $N$ boxes appears. Determine the number of valid arrangements satisfying this condition. Since