Permutations problem

derb12

Can anyone help with this problem.


n identical eggs are to be disributed amongst 8 people.

Each person will get at least one egg.
The total number of eggs may be arbitrarily large.


We have to show that there are
(n-1)
(8-1)
configurations such that n eggs are distributed.


I can see how this works in practice and that the answer will always be
(n) - (n-1)

(8) (8)


but I can't get the algebra to work.

* That should be one big bracket around the expression above as in

[(n-1) choose (8-1)]


Thanks in advance for any help with this question or pointers to online resources that explain it.

MathsManiac

This is a variant on the problem of placing n indistinguishable objects into r (distinguishable) boxes, which you can Google if you wish.

Here, the variation is that everyone gets at least one egg. You can get back to the 'standard' version of the problem by dealing with this aspect first. Give everyone one egg. Now the problem is how to distribute the remaining n-8 eggs among 8 people, without further restriction.

You will find a treatment of the problem of distributing M indistinguishable objects into N distinguishable boxes in various places on the web, including here:
https://math.stackexchange.com/questions/2522240/m-indistinguishable-balls-on-n-indistinguishable-boxes/2522265

Hope this helps.

derb12

Thanks for that pointer.


Just for posterity, here is the easiest way I found of thinking about it and no algebra required!


Imagine that rather than giving eggs to people, we just line them up in a row and divide the row into 8 sections using partitions.



There will be n eggs and therefore n-1 gaps between the eggs where you can place the partitions.


As we need 8 sections we need 8-1 partitions.


Then we simply need to know how many ways can 8-1 partitions be placed into n-1 slots ie [(n-1) choose (8-1)].