Since we can map such sequence 

$$0\leq a_1\leq a_2\leq a_3 \leq \cdots \leq a_{n-1}\leq a_n < m   $$
to
$$0 < b1=a_1+1 < b2=a_2+2 <  b3=a_3+3 <\cdots  < b_n=a_n+n < m+n  $$
and
$\{b_1,b_2,\cdots b_n\}$ is  the `n` subsets of `Range[m+n]`

And we can get $\{a_1,a_2,\cdots a_n\}$ from $\{b_1,b_2,\cdots b_n\}-\{1,2,\cdots,n\}$

```
With[{m = 10, n = 5}, 
 Subtract[Range[n], #] & /@ Subsets[Range[m + n], {n}]]

```