Counting

First lay out the number of ways to distribute the 4 balls: 

 

4 - 0 - 0 - 0 - 0

3 - 1 - 0 - 0 - 0

2 - 1 - 1 - 0 - 0

2 - 2 - 0 - 0 - 0

1 - 1 - 1 - 1 - 0

 

For the first case, there are \({5 \choose 1} = 5\) ways to choose the box for the 4, and the rest must be 0

 

For the second case, there are \({5 \choose 3} \times {2 \choose 1} = 20\) ways.

 

For the third case, there are \({5 \choose 1} \times {4 \choose 2} = 30\) cases

 

For the fourth case, there are \({5 \choose 2} = 10\) cases. 

 

For, the final one, there are \({5 \choose 1} = 5\) ways. 

 

So, in total there are \(5 + 5 + 10 + 20 + 30 = \color{brown}\boxed{70}\) ways.