Combinatorics

Stars and bars

Share nn things among kk boxes, with a minimum per box.

For checking your work after you’ve tried it yourself.

Step 1:
Type n and k

n identical things, k different boxes.

Step 2:
Set a minimum

The same for every box, or one per box.

Step 3:
Read the count

And see every way when there are only a few.
What it does.
Counts the ways to share nn identical things among kk different boxes, optionally with a minimum in each box.
When it helps.
Checking a sharing-out count you worked out.
What it won’t do.
The things must be identical and the boxes different. It doesn't handle maximums.
Example.
7 sweets for 3 children, at least 1 each: (62)=15\binom{6}{2} = 15 ways.
Share identical items among boxes

n up to 1,000,000; k up to 1000. Leave the minimum empty for 0.

Try