MHB A simple (?) combinatorial problem

[lfdahl]

As simple as it seems, I still can´t figure out how to solve the following combinatorial problem:

In how many ways can 20 identical items be distributed among 10 (different) persons, when each person may have from zero to five items?

Thankyou in advance for any help, that can lead me on the right path ...

 

[I like Serena]

As simple as it seems, I still can´t figure out how to solve the following combinatorial problem:

In how many ways can 20 identical items be distributed among 10 (different) persons, when each person may have from zero to five items?

Thankyou in advance for any help, that can lead me on the right path ...

Hey lfdahl,

It would be simple if we could apply the Stars and bars method.
But since we have a maximum of 5 items per person, I believe it's not so simple.
Simplest that I can come up with, is to programmatically enumerate all possibilities.
That should be doable since we have an upper boundary of $6^{10}$, which is still feasible.

In python:

""" Number of ways to divide o identical objects over p persons
where each person receives 0 to m objects.
"""
def recurse(o, p, m):
	if p == 1:
		return 1 if o <= m else 0
	n = 0
	for i in range(0, min(m, o) + 1):
		n += recurse(o - i, p - 1, m);
	return n;
	
print recurse(20, 10, 5);

Result is [M]2930455[/M].

 

[lfdahl]

Hey lfdahl,

It would be simple if we could apply the Stars and bars method.
But since we have a maximum of 5 items per person, I believe it's not so simple.
Simplest that I can come up with, is to programmatically enumerate all possibilities.
That should be doable since we have an upper boundary of $6^{10}$, which is still feasible.

In python:

""" Number of ways to divide o identical objects over p persons
where each person receives 0 to m objects.
"""
def recurse(o, p, m):
	if p == 1:
		return 1 if o <= m else 0
	n = 0
	for i in range(0, min(m, o) + 1):
		n += recurse(o - i, p - 1, m);
	return n;
	
print recurse(20, 10, 5);

Result is [M]2930455[/M].

Hey, I like Serena

Thanks a lot for your thorough answer to my problem!
Although I´m not familiar with python, I will see, if I can write your codes in SciLab.
I really appreciate your help!