Positive solutions with lower bounds for JEE

Formula

\(x_1+x_2+\cdots+x_n=r,\ x_i\ge 1\Rightarrow\) number of solutions \(=\,{}^{r-1}C_{n-1}\)

Variables: \(x_i\) are positive integers.

Conditions: \(r\ge n\).

At least one item per box, partitioning totals into positive parts.