Non-negative solutions with lower bounds for JEE

Formula

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

Variables: \(a_i\) are given non-negative integers; set \(y_i=x_i-a_i\).

Conditions: \(r\ge \sum a_i\).

Integer solution problems with minimum allocation constraints.