How many strictly positive integer solutions does the equation $x_1+x_2+···+x_n = k$ have? (Hint: Consider the equation $y_1+y_2+· · ·+y_n = k−n$ with variables $y_i \ge 0$.)
I believe the second equation represents the first, but fixes $k$ as a positive integer. How can I show this rigorously?
In turn, this means there are $n-1+k-n$ total elements and $n-1$ elements chosen. I will then compute the binomial coefficient with these values. Is this correct?