Number of combinations with limited repetition


by EmileMaras
Tags: combinations, combinatorics, limited, number, repetition
EmileMaras
EmileMaras is offline
#1
Dec12-13, 06:00 AM
P: 2
Hello

I have the following combinatoric problem :
I want to distribute n (equivalent) atoms among M distinct objects. Each object can contain from 0 to nlim atoms. How many combination do I have for this system?

If nlim>n, this problem actually corresponds to the classical "Number of combinations with repetition". But in my case nlim<n. In fact, I am interested in the limit of (lnΩ)/n (Ω beeing the number of combination) when M and n tend toward infinity (with n=a M where a is a constant) while nlim is finite (and actually rather small)..

I found a solution for that problem using some series of summations but it will be impossible to caculate as soon as M and n become large (even for M=100, n=300 and nmax=10, it took my laptop more than one hour to solve it).
Is there a simple analytical solution to this problem?

Thank you for your help.

Emile Maras
Phys.Org News Partner Mathematics news on Phys.org
Researchers help Boston Marathon organizers plan for 2014 race
'Math detective' analyzes odds for suspicious lottery wins
Pseudo-mathematics and financial charlatanism
Shyan
Shyan is offline
#2
Dec12-13, 06:37 AM
Shyan's Avatar
P: 734
For the first atom,there is M choices...for the second,again M choices...for the third,again M choices...and so on!
So there is always M possible choices and all that we should do is to multiply the number of choices for each of the atoms which becomes [itex] M^n [/itex]. But because the atoms are identical,we should decrease this amount by dividing it by [itex] n! [/itex].
EmileMaras
EmileMaras is offline
#3
Dec12-13, 08:32 AM
P: 2
I guess that it is not the correct answer. Maybe I did not state my problem properly, so I will just give an exemple.
Let's say I have n=3 atoms and M=3 object. An object can contain at max nmax=2 atoms. Then the possible combinations are 111, 012, 021, 102, 120, 201, 210 (where xyz gives the number of atom in each object) which corresponds to 7 combinations.

Shyan
Shyan is offline
#4
Dec12-13, 08:46 AM
Shyan's Avatar
P: 734

Number of combinations with limited repetition


Quote Quote by EmileMaras View Post
I guess that it is not the correct answer. Maybe I did not state my problem properly, so I will just give an exemple.
Let's say I have n=3 atoms and M=3 object. An object can contain at max nmax=2 atoms. Then the possible combinations are 111, 012, 021, 102, 120, 201, 210 (where xyz gives the number of atom in each object) which corresponds to 7 combinations.
Yeah,my answer is wrong.It even gives a non-integral value!
Anyway,Check here!


Register to reply

Related Discussions
Combinations with Irregular Repetition Set Theory, Logic, Probability, Statistics 3
Calculating number of possible combinations with limited repitition Set Theory, Logic, Probability, Statistics 16
Combinations with limited repitition General Math 10
Does Ohm's Law still apply when the total number of Amps is limited? General Physics 2
Combinations with repetition (an intuitive way to it?) Set Theory, Logic, Probability, Statistics 2