Distributing r identical obejcts into n distinct containers.

Question

So, lets say I have $4$ burglars, and I have a custom pistol with $10$ bullets in it. How many possible ways can I put $10$ bullets into EXACTLY $2$ burglars and let the other $2$ get away?

Note that I am NOT looking for an ANSWER, I am only asking if it is possible for someone to help me understand why my initial conditions are incorrect. In particular, can someone confirm if what I am doing below is correct?

*My technique is to use the inclusion exclusion formula where it is

$$E_2 = S_2 - \binom{3}{1}S_3 + \binom{4}{2}S_4$$

where $S_i$ stands for the possibilities where $i$ burglars get a bullet in them.

Now, lets say I want to find $S_2$, which is the number of ways to distribute $10$ bullets amongst $2$ burglars. Would the answer to this perhaps be

$$\binom{4}{2}(10^2 - 2)$$

where we first choose $2$ burglars out of the $4$ and then with the $2$ burglars each can have $10$ bullets so $10^2$ and then I minus $2$ because each has to have at least $1$.

Can someone please confirm if I did $S_2$ correctly?

The answer is $6132$ just in case anyone is wondering...

Please see this tutorial on how to typeset mathematics on this site. Also, if you right click on an expression, you can see how I formatted it by checking the LaTeX commands. — N. F. Taussig, Apr 07 '15 at 09:10

David · Accepted Answer · 2015-04-07T03:46:33.287

2

As far as I can see, your explanation for $S_2$ is correct, but in your answer you wrote $10^2$ when it should be $2^{10}$.

Also, it seems to me that this should be the entire answer, you don't need to consider $S_3$ and $S_4$.

edited Apr 07 '15 at 03:46

answered Apr 07 '15 at 03:41

David

82,662

Why is it 2^10..? I am confused..? We only have 2 burlagrs so shouldn't it be 10x10 sir? – Belphegor Apr 07 '15 at 03:42
If you had two burglars and one bullet, how many ways are there to use the bullet? Is it $2^1$ or $1^2$? – David Apr 07 '15 at 03:43
It would seem that it is indeed the answer.. O.O, but the hint that was given to me wishes me to use the formula where S3 and S4 are involved. Would the answer follow if I use the logic in S2 and apply to S3 and S4? – Belphegor Apr 07 '15 at 03:46
Sorry but I have no idea what that hint means. – David Apr 07 '15 at 03:47
I see, no problem, it is just a corollary that allows one to find the "exact" number of m conditions that satisfy t conditions. – Belphegor Apr 07 '15 at 03:49
Anyways, thank you for correcting my work! :D – Belphegor Apr 07 '15 at 03:51
You're welcome! – David Apr 07 '15 at 03:52

Distributing r identical obejcts into n distinct containers.

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