What is the total number of solutions of an equation of the form $x_1 + x_2 + \cdots + x_r = m$ such that $1 \le x_1 < x_2 < \cdots < x_r < N$ where $N$ is some natural number and $x_1, x_2, \cdots, x_r, m$ are integers?
Also, what would be the solution in the relaxed case where $x_1, x_2, x_3, \dots, x_r$ could be equal?