possible number of sums

Question

I need help finding a general formula for this specific problem-

$$(a_1+a_2+a_3+...+a_n)= x, \ \ \ a_1,a_2,a_3,...,a_n∈[0,x] $$

How many permutations/combinations of this sum can be formed? (all positive integers and repetition allowed)

Here's an example where $x=2$ and $n=3$.

$$a_1+a_2+a_3=2, \ \ \ \ a_1,a_2,a_3\in[0,2]$$

$0+0+2 ...(1)$

$0+2+0 ...(2)$

$2+0+0 ...(3)$

$1+1+0 ...(4)$

$1+0+1 ...(5)$

$0+1+1 ...(6)$

Hence total $6$ combinations, but I can't seem to figure out a general formula for this.

By the way, this is my first question here so please excuse my unprofessional questioning.

Answer 1

This is just stars and bars concept.stars and bars

If you want a formula then okay. Here it is

$\binom{n+x-1}{x}$