Question #74703

Let $A$ and $B$ be two distinct points in $\mathbb{R}^3$. For $i = 1, 2, \dots, n$, let $P_i$ and $Q_i$ be the points dividing the line segment $AB$ internally and externally in the ratio $\lambda_i : 1$ respectively, where $\lambda_1, \dots, \lambda_n$ are distinct positive real numbers not equal to 1. Let $S = \{P_1, \dots, P_n, Q_1, \dots, Q_n\}$. Let $N_k$ be the number of bijections $f: S \to S$ satisfying the following conditions: (I) $f$ maps every internal division point to an external division point, and every external division point to an internal division point. (II) Exactly $k$ internal division points map to their corresponding external division points (i.e., $f(P_i) = Q_i$ for exactly $k$ indices). (III) No external division point maps to its corresponding internal division point (i.e., $f(Q_i) \neq P_i$ for all $i$). If $D_m$ denotes the number of derangements of $m$ distinct objects, which of the following represents the value of $\sum_{k=0}^{n} N_k$?