Balls Into Boxes, Empty Allowed

This is a stars-and-bars problem with identical balls and labeled boxes.

Use the formula $\left(\genfrac{}{}{0ex}{}{n+k-1}{k-1}\right)$ for $n$ balls and $k$ boxes.

1. 1. Apply stars and bars

   With $n=10$ balls and $k=4$ boxes, the count of nonnegative integer solutions is $\left(\genfrac{}{}{0ex}{}{n+k-1}{k-1}\right)$.

   $$\left(\genfrac{}{}{0ex}{}{10+4-1}{4-1}\right)=\left(\genfrac{}{}{0ex}{}{13}{3}\right)=286$$

Answer: 286