Positive Integer Solutions to Equation

Rewrite equation as $y + z = x - 25$. Since $x \le 40$, $y + z \le 15$. Since $x \ge 1, y \ge 1, z \ge 1$, and $x = y + z + 25$, for each valid pair $(y, z)$ with $1 \le y, z \le 12$ and $y + z + 25 \le 40 \implies y + z \le 15$. We need number of pairs $(y, z)$ of positive integers such that $y + z \le 15$ and $y, z \le 12$. For $k = y + z$ where $2 \le k \le 15$:

• $k = 2$: 1 solution
• $k = 3$: 2 solutions ...
• $k = 13$: $(1,12)$ to $(12,1) = 12$ solutions
• $k = 14$: $(2,12)$ to $(12,2) = 11$ solutions
• $k = 15$: $(3,12)$ to $(12,3) = 10$ solutions Sum from $k=2$ to $13$: $1 + 2 + \dots + 12 = 78$. Plus $k=14$ (11) and $k=15$ (10): $78 + 11 + 10 = 99$.