Solving for Points in 4D Space with Nonnegative Integer Coordinates

[mathmajor23]

Homework Statement

How many points (x1,x2,x3,x4) in the 4-dimensional space with nonnegative integer coordinates satisfy the equation x1 + x2 + x3 + x4 = 10?

I'm not sure which method to use to start this problem. Any ideas?

 

[jedishrfu]

forget the 4D for the moment and consider all the ways you can add up numbers to get 10:

0+0+0+10 = 10
0+0+1+9 = 10
1+1+1+7 = 10
...

and next you consider for each 4 number sum and ask yourself how many 4D points can have coordinate points of 1,1,1,7:
(1,1,1,7) (1,1,7,1) (1,7,1,1) (7,1,1,1)

 

[mathmajor23]

That will take forever to use brute force.

Thinking combinatorially, my initial thought would be C(10+4-1,3) = C(13,3) = 286 different ways. Any thoughts?

 

[morphism]

That will take forever to use brute force.

Thinking combinatorially, my initial thought would be C(10+4-1,3) = C(13,3) = 286 different ways. Any thoughts?

That's correct.

More generally, can you show that the number of solutions to $x_1 + x_2 + \cdots + x_k = n$ with each x_i a nonnegative integer is C(n+k-1,k-1)?