Question #40579

This question tests the intersection of combinatorics (Stars and Bars) with algebraic inequalities.

The core trap lies in the 'Cross Multiplication Trap' (Concept 62).

Students often blindly cross-multiply $\frac{2}{k-4} \ge \frac{1}{k-9}$ to get $2k-18 \ge k-4 \implies k \ge 14$, completely missing the region where the product of the denominators is negative ($4 < k < 9$).

This leads them to calculate only the solutions for $k \in \{14, 15\}$, yielding the trap answer 256.

A rigorous approach requires bringing all terms to one side and using the wavy curve method to identify all valid bounded regions before applying the $\binom{n+r-1}{r-1}$ distribution formula.