Boxes With Minimums

Give each box 2 balls first, then distribute what remains freely.

Use stars and bars on the reduced problem.

1. 1. Reduce the problem

   Give each of the 5 boxes 2 balls upfront, using $10$ balls; $5$ balls remain to distribute freely.

2. 2. Apply stars and bars

   The number of ways to distribute 5 identical balls into 5 labeled boxes with no restriction is $\left(\genfrac{}{}{0ex}{}{5+5-1}{5-1}\right)=\left(\genfrac{}{}{0ex}{}{9}{4}\right)$.

   $$\left(\genfrac{}{}{0ex}{}{9}{4}\right)=126$$

Answer: 126