<s>[INST] Combinatorics problem ascending order numbers Let $n_1 < n_2 < n_3 < n_4 < n_5$ be positive integers such that $n_1 + n_2 + n_3 + n_4 + n_5 = 20$. Then the number of such distinct arrangements $(n_1, n_2, n_3, n_4, n_5)$ is ....
 [/INST] The solutions must be of the form 
$$(n_1,n_2,n_3,n_4,n_4)=(1+a,2+b,3+c,4+d,5+e)$$
with
$$0\le a\le b\le c\le d\le e\quad\text{and}\quad a+b+c+d+e=5$$
These correspond to the partitions of $5$ (fleshed out with $0$'s), of which there are $7$, namely
$$\begin{align}
0+0+0+0+5\\
0+0+0+1+4\\
0+0+0+2+3\\
0+0+1+1+3\\
0+0+1+2+2\\
0+1+1+1+2\\
1+1+1+1+1\\
\end{align}$$
</s>