There are twenty-five identical marbles to be divided among four broth

Hi firas92 can you further elaborate on the below point.

"Now the remaining 13 marbles must be distributed in such a way that any one can get any share

Consider the 13 marbles being separated by 3 separators such that there are four slots in all (representing the four brothers)

*************|||

We can arrange these 16 elements in \(\frac{16!}{13!3!}\) ways = 560 ways"

Thanks.

Hi abhishek31

That is just a visualization of the formula \((n+r−1)C(r−1)\) where \(n\) identical items are distributed among \(r\) participants such that any participant may receive any number of items.

In our case, the constraint we are given is that each brother must have at least 3 marbles. How the distribution proceeds after that is not constrained in any way. Which means, once each brother has 3 marbles, we can distribute the remaining in any ratio (13-0-0-0) or (10-1-1-1) or (5-5-2-1) and so on. And remember that there is only 1 way in which we can distribute 3 marbles to each brother because the marbles are all identical.

Now I have represented the 13 marbles as *************

Now let's create 4 slots using separators ||| (each slot representing one brother) - we have to use 3 separators to make 4 slots

____Slot 1___ | ___Slot 2___ | ___Slot 3___ | ___Slot 4___

The way in which the separators are inserted determines the share. (For example *************||| = 13,0,0,0; ****|****|**|*** = 4,4,2,3; ******|*******|| = 6,7,0,0; and so on)

The above are just 3 examples of many cases. So how many ways are there to insert the separators? To calculate this, we just have to find the number of permutations involving 13 identical marbles and 3 identical separators which is given by \(\frac{16!}{13!3!}=560ways\)

Hope this is clear!

To more people like me who were wondering what are the stars and bars, here is a complete walk through.
https://youtu.be/40HxI6Uc00Q

All the best!

Hey Bunuel can you please provide your 2 cents on this problem. I was thinking along the lines :- first divide 3 marbles to each 4 brother, then divide the remaining 13. Now each marble has 4 choices i.e 1st,2nd,3rd, or 4th brother. So \(4^{13}\). But obviously its wrong, Can you please help me understand how am I supposed to think in such questions. Also if you could redirect me to similar questions, that would be really helpful.

Thanks Bunuel. It helped a lot.