Question #40611

This question tests the synthesis of combinatorics (dictionary rank with repeating letters) and calculus (indeterminate forms requiring Taylor series expansion).

The trap lies in two places: First, students often miscalculate the rank by mishandling the permutations of identical letters (e.g., forgetting to divide by $2!$ for 'E' or 'I', leading to $R=192$ and the distractor $e^{32}$).

Second, the limit requires expanding trigonometric functions to the third power ($x^3$); using only the first-order approximation ($\sin x \approx x, \tan x \approx x$) leads to a $1^\infty$ form that incorrectly evaluates to $e^0 = 1$, forcing them to re-evaluate using higher-order terms or L'Hôpital's rule.