Combinatorics: Distributing Identical Walls into Multiple Boxes with Constraints

[saint_n]

i need help with combinatorics...i need to finda ogf to compute the how many ways can 27 identical walls be distributed into 7 boxes, where the first box can contain at most 9 balls

How do this??can you give me a method or explain to me how to do this step by step PLEASE!

sAint

 

[matt grime]

i think you ought to at least start trying to count the ways of doing this yourself. because that's all you're doing. there is no clever trick necessary, just count them, how many can you put in the first one, how many in the second, and so on. do you count putting no balls in a box as being permissible. think about smaller examples first, say 3 boxes 6 balls, no more than two in the first, or even smaller than that so you get the idea.

just to reiterate, the solution is to just count the ways, and that's something you can do if you don't presume it's beyond your capabilities.

 

[Gokul43201]

Imagine the balls were laid out in a straight line. What happens when you insert a wall at some point between 2 of the balls ? Is this at all like putting the balls into boxes ? How many walls do you need, to simulate 2 boxes, or 3, or 4, ...or n ? How many ways are there of inserting these walls ?

 

[saint_n]

Sorrie haven't replied for a while been kinda busy and sorrie again i thought there was a method because in our notes(which doesn't really help much) he has an example
Suppose choose 25 objects from 7 types of objects s.t. every type of object appears at least twice and at most 6 times..
he wrote some formula
[x^25](x^2 + x^3 +x^4 +x^5 +x^6)^7

so i thought x^25 is the 25 objects and the exponents 2 to 6 is the "at least twice and at most 6 times" and ^7 is the seven types of objects then he did some calculations which i understand and he got 6055.

I just got confused when the question i asked you before because i didnt know how to represent the way he did but I am not sure since you were helping me out could my answer for the prev. question be
[x^27](1 + x + x^2 + ... + x^9)(1 + x + .. + x^27)^6

am i on the rite track?

 

[saint_n]

btw can you recommend a combinatorics book because he's notes help only a lil plus sometimes he doesn't explain himself properly(by the way we don't have a prescribed textbook) so anything we would be appreciated...

 

[matt grime]

what you have written there as a polynomial is called a generating function, these are very useful and powerful tools. what is important is the coefficients of the powers of x.

For instance, fix n, then the number of ways of choosing r objects from n (order unimportant) is the coefficient of x^r in the expansion of (1+x)^n

 

[saint_n]

ok,,so what i have now is correct?Just so i can get started...