Arrange balls in bins

Hi,

I am Aman Ratan, a mechanical engineer and i am currently involved in beam calculation for more than 5 supports.

Its an uncharted territory so to speak.

The problem statement is as follows ---



I want to distribute 'N' number of units (each with value 10) into '4' groups and then obtain the sum of those groups individually in the form of a list (All Possible combinations and not permutations).



Its easier if i explain it with an example,

Suppose the number is 1000

dividing it into 100 parts, we get 100 units each with value 10.

now lets consider combinations in which these 100 units can be distributed into as 4 sets.

i.e. 97 - 1 - 1 - 1
next can be 96 - 2 - 1 - 1
then 95 - 3 - 1 - 1

and so on till 1 - 1 - 1 - 97


The aim is to develop a program that can list all the possible combinations like above (not permutations)

Sorry for the inability to convey the problem statement properly.

Any help will be appreciated.

*Admins, not sure if i have posted the problem in the appropriate thread. If not please make neccessary changes at your discretion.

97 - 1 - 1 - 1
...
and so on till 1 - 1 - 1 - 97
...
(not permutations)

1-1-1-97 is a permutation of 97-1-1-1 so this routine would return only one of them, not both.

Perhaps this will do what you want.
I also think that it might be wise for you (or one of the mods) to remove the email address from your post.
Note that the shape of ElementsSummingTo(100,4) is 4 rows by 7153 columns.

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amanratan welcome to the forum

Please take a moment to read the forum rules and then amend your thread title to something descriptive of your problem - not what you think the answer might be. (think google search terms?). Once you have done this please send me a PM and I will remove this request. (Also, include a link to your thread - copy from the address bar)

Many members search our previous posts, and thread titles play a big part of the search. I doubt anybody would do a search based on your title?

To change a Title on your post, click EDIT POST then Go Advanced and change your title, if 2 days have passed ask a moderator to do it for you.

(note: this change is not optional )

I would understand your problem as, "list the ways in which I can arrange N balls into M distinct bins, with no bins empty", which is =COMBIN(N-1, M-1)

E.g., for N=100 and M=4, =combin(100-1, 4-1) = 156849:

Row\Col	A	B	C	D
1	1	1	1	97
2	1	1	2	96
3	1	1	3	95
4	1	1	4	94
5	1	1	5	93
6	1	1	6	92
156843	95	2	1	2
156844	95	2	2	1
156845	95	3	1	1
156846	96	1	1	2
156847	96	1	2	1
156848	96	2	1	1
156849	97	1	1	1

Ensemble, those are not combinations, permutations, or partitions, just particular arrangments.

If that's it, I suggest you change your thread title to "Arrange balls in bins"

Row\Col	B	C	D	E	F
2	Balls	Bins
3	20	4
4	# Arrangements
5	969
6
7	Arr #	Bin 1	Bin 2	Bin 3	Bin 4
8	1	17	1	1	1
9	2	16	2	1	1
10	3	16	1	2	1
11	4	16	1	1	2
12	5	15	3	1	1
13	6	15	2	2	1
14	7	15	2	1	2
15	8	15	1	3	1
16	9	15	1	2	2
17	10	15	1	1	3
18	11	14	4	1	1
19	12	14	3	2	1
20	13	14	3	1	2

See the workbook at https://app.box.com/s/d6mo50rxjiipx01xj9sv15soh8rna8nd

The genesis of the formula is this:

Imagine N balls in a row. You can insert a divider between any two adjacent balls to partition them into bins. To create M bins, you need M-1 dividers, and there are N-1 slots to choose from. So =COMBIN(N-1, M-1) gives the number of arrangements.

I'm not sure what the repeated injunction "not permuations" means in this context.
Shg's solution has what some might call permutations, but the so does the OP questions.

I think about permutations as ordered rearrangements of a set (or a subset of a set). I can't see what set this list is the permutations of.