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
