How many combinations are there to put n chapters into k（<n）volumes?

Also, how can algorithmicly generate them within polynomial of n? Pseudo-code is ok.

This problem comes down to selecting k-1 borders, which divide n elements in k parts. As such, k-1 positions need to be selected out of n+k-1 positions, resulting in

possibilities. As an example, let's say we need to divide four sweets among three children. In this case, a first possibility would be to put the two borders on the first two positions, leaving the four sweets in positions 3-6:

As such, the first two children get no candy, while the third child gets all four. Another possibility would be to put the first border at position 2 and the second border at position 4:

Now the first two children get one candy each (positions 1 and 3), while the third child gets two (positions 5-6). Iterating over all positions results in a total of 15 possibilities.

Note that the answer is different when all volumes need to contain at least one chapter. In this case, k items are already given to one volume each, leaving n-k chapters. As such, there are

possibilities. Note that the condition k<n is important in this case.