You will never understand this question unless you yourself sit and work with the numbers.
May I suggest that you cut the question down to 100 balls an let K=8.
List the smallest eight numbers. What is
List the larest eight numbers. What is
List any eight numbers. What is
Now what is the maximum for ?
Now what is the minimum for ?
Now what is ?
The think about number partitions?
When you have done that. Tell us what you find.