Find permutation of N numbers which gives the sum of X

Question

Assume

  n = 3  //A,B,C
  x = 6

How to find all permutaions of three numbers which makes to the total of 6. For example how to calculate how many permutations we can find.

A   B   C
----------
0   0   6
0   6   0
6   0   0

0   1   5
1   0   5
0   5   1
1   5   0
5   0   1
5   1   0

0   2   4
1   1   4
2   0   4
0   4   2
2   4   0
1   4   1
4   1   1
4   0   2
4   2   0

3   0   3
3   3   0
0   3   3

3   2   1
3   1   2
1   3   2
2   3   1
1   2   3
2   1   3

2   2   2

What is the formula to count the purmutations for given x and n, to get results like above? Is there any programatical way to do this

Answer 1

First, find how many different groups of numbers sum to $6$, and write them as an unordered set. $\left \{ 1,2,3 \right \},\left \{6,0,0 \right \},\left \{5,1,0 \right \},\left \{2,4,0 \right \}$
Now find how many ways you can "shuffle" or arrange the numbers in each of these sets. Each set has $3$ elements, that means there are $3!$ ways to arrange each set. There are $4$ sets, so; $3!\times4=28$