Hello,

the task is to calculate, how many solutions $\displaystyle (x_1, x_2,..., x_n) \in \mathbb{Z}^n$there is to equation $\displaystyle x_1+x_2+...+x_n=k$. Every solution must be $\displaystyle x_i \geq i$ for all $\displaystyle i=1,2,...,n$.

I got something like this:

$\displaystyle y_i=x_i-i \geq 0$

$\displaystyle (x_1-1)+(x_2-2)+...+(x_n-n)=k-\frac{n(n+1)}{2}$

$\displaystyle y_1+y_2+...+y_n=k-\frac{n(n+1)}{2}$

$\displaystyle \Rightarrow \left(\begin{array}{cc}\frac{n(n+1)}{2}+k-1\\k\end{array}\right)$

Is that right or Am I missing something? Any help is appreciated. Thank you.