Number of ways to split b balls into n bins is:

Why? Think about this diagram ***|**||*** for dividing 8 balls into 4 bins as (3,2,0,3)

Number of ways to split b balls into n bins with at least one ball in each bin:

Why? Think about this diagram o*|o**|o*|o** for dividing 10 balls into 4 bins as (2,3,2,3)

If you want to calculate exactly x bins filled, then this is easy

Why? Pick x bins and fill at least 1 balls in each bin.

If you want at least x bins filled, then it is probably easiest to do a summation: