{Number theory} Integer solutions

[youngstudent16]

Homework Statement

$x_1+x_2 \cdots x_{251}=708$ has a certain # of solutions in positive integers $x_1 \cdots x_{251}$
Now the equation $y_1+y_2 \cdots y_{n}=708$ also has the same number of positive integer solutions $y_1, \cdots y_n$ Where $n \neq251$ What is $n$

Homework Equations

I think this is a stars and bars problem but I'm not super familiar with it still

The Attempt at a Solution

So looking at the stars and bars page it seems that ${m \choose k}={m \choose m-k}$ so then would $n$ in this case just be $457$? In this case $m=707$ and $k=250$

Edit figured it out it is $458$ I just forgot to add 1 back to the original $n$

 

[HallsofIvy]

I don't think I understand the notation here. You say this problem "has a certain # of solutions in positive integers x ₁ ⋯x₂₅₁" I would take that to mean that it has 251 solutions. You then say "y₁+y₂ \cdots y_n=708 also has the same number of positive integer solutions y1 ,⋯yn y_1, \cdots y_n Where n≠251".

If the first equation has 251 solutions and the next has "the same number" how is it not 251?

And I have no idea what a "stars and bars problem" and a "stars and bars page" are!

 

[haruspex]

And I have no idea what a "stars and bars problem" and a "stars and bars page" are!

I believe it refers to the problem of how many ways of placing r identical objects into n distinct buckets. Maybe the 'bars' represent the divisions between the buckets.
The posted solution, after correction, looks right.

 

[pbuk]

I don't think I understand the notation here. You say this problem "has a certain # of solutions in positive integers x ₁ ⋯x₂₅₁" I would take that to mean that it has 251 solutions.

No, the OP simply means that it "has j solutions", where j is unknown.

And I have no idea what a "stars and bars problem" and a "stars and bars page" are!

This refers to an interesting class of problems with an elegant path to solution; there are pages that explain this further on both Mathworld and Wikipedia which can be found using a well-known search engine.

Is the question asking for the number of unique solutions, or are permutations of $x_i$ permitted?

 

[pbuk]

... and have you dealt properly with solutions ending with a bar (that will not sum to 708) and solutions starting with one or more stars that will not satisfy that $x_i$ are positive?

Well done for spotting the stars and bars analogue though, it is not immediately obvious.