Math Problem Statement

Five people are in a group, including Kai and Wen. To practice social distancing, they first take the step of splitting the group into two disjoint subgroups. Then, they undergo a series of additional steps, during each of which they divide every remaining subgroup that has more than 1 person into two further subgroups. The steps end when every person is alone. Due to a longstanding grudge, Kai and Wen cannot be together in the same subgroup after the first division. Considering that the order of divisions matters and all remaining subgroups are divided simultaneously at each step, how many possible division pathways exist? A division pathway is considered the full process starting from 1 group of 5 and ending at 5 groups of 1.

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Solution

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Math Problem Analysis

Mathematical Concepts

Combinatorics
Symmetry in Combinatorial Problems

Formulas

Binomial Coefficient \( \binom{n}{k} \)

Theorems

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Suitable Grade Level

Grades 10-12