The question is, how many $k-$dimensional non-negative integer arrays $(x_1,\cdots,x_k)$ satisfies $x_1+x_2+\cdots+x_k\le n$?
For example for $k=3,n=2$ we have $(0,0,0),(1,0,0),(0,1,0),(0,0,1),(2,0,0),(1,1,0),(1,0,1),(0,2,0),(0,1,1),(0,0,2)$ overall $10$ such vectors. Can we obtain a formula for this?
The number definitely smaller than $n^k$ due to the constraint. But how much is it smaller?