Question

How many non-negative integer solutions does this equation have?
\[x_1+x_2+\ldots+x_N=n\]

Answer 1

Are you considering the solutions to be ordered or unordered?
For example, if we wee trying to solve:
x + y + z + w = 12
Would the solution (x,y,z,w) = (9,1,1,1) = (1,9,1,1) = (1,1,9,1) = (1,1,1,9)
or would that be considered 4 separate solutions?
If they are considered the same, then this is a partitions problem, if they are different, then this is a combinations problem.

Answer 2

They are different.

Answer 3

woops!! that's for ... a+b+c+d = N

Answer 4

For the smaller example: x + y + z + w = 12

The number of solutions can be determined by viewing the problem in the following way:
If 12 units (denoted as 12 u's) seperated by 11 spaces (denoted by 11 s's) are lined up,

usususususususususususu

and we choose any 3 of the s's and let the others disappear, for example:
u u s u u u usu u u u usu

then the remaining s's seperate the units into four batches. The number of units in these batches can be used as the values of x,y,z, and w.

Answer 5

a+b = n would make n+1 solutions
a +b + c = n would make