Balls Into Boxes, None Empty

Place one ball in each box first to satisfy the minimum requirement.

Distribute the remaining balls freely using stars and bars.

1. 1. Reduce to the unrestricted problem

   After giving each of the $4$ boxes one ball, $6$ balls remain to be freely distributed.

   $$\left(\genfrac{}{}{0ex}{}{6+4-1}{4-1}\right)=\left(\genfrac{}{}{0ex}{}{9}{3}\right)=84$$

Answer: 84