If the smallest of the three groups contains
only 1 ball, the other two groups must contain
the other 11 balls, which can only be done with
the middle sized group having 5 and the largest
group having 6.

1.  The 3 groups are of sizes 1, 5,  and 6. 


If the smallest of the three groups contains
2 balls, the other two groups must contain
the other 10 balls, which can be done two
ways. The middle sized group can have 4 and the 
largest group has 6.  Or they can have 5 each:

2.  The 3 groups are of sizes 2, 4  and 6.
3.  The 3 groups are of sizes 2, 5  and 5.

If the smallest of the three groups contains
3 balls, the other two groups must contain
the other 9 balls, which can be done two
ways. The second group can also have 3 and the 
largest group 6.  Or the middle sized 
group can have 4 and the largest group 5.

4.  The 3 groups are of sizes 3, 3  and 6.
5.  The 3 groups are of sizes 3, 4  and 5. 

One more way.  They can have 4 balls each:

6. The 3 groups are of sizes 4, 4, and 4.


So there are 6 ways to divide the 12 balls into 3 groups,
with no group larger than 6.

Edwin