Alper is the owner of Cube-XYZ, a perfect cube-shaped planet, and builds a mansion on one vertex of the cube, and distinct houses on the seven other vertices. Houses are considered next door if they are connected by an edge of the cube. He is to host an intergalactic tea party for his friends; Wes, Ethan, Henry, Alan, James, Vincent, and Tom.  They are to be accommodated in separate houses. As host, Alper has conditions on how the guests must be placed on his planet:   i. His best friend Ethan must live next door to him ii. Henry and James must not live next door to each other iii. Alan and Vincent must live next door to each other iv. Wes should be as many edges away from Alper as possible v. Alper lives in the Mansion vi. Tom lives (strictly) closer in distance to Wes than Alper   i. Given all conditions are met, how many combinations of living arrangements are there?

Algebra and Trigonometry (6th Edition)
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ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Alper is the owner of Cube-XYZ, a perfect cube-shaped planet, and builds a mansion on one vertex of the cube, and distinct houses on the seven other vertices. Houses are considered next door if they are connected by an edge of the cube. He is to host an intergalactic tea party for his friends; Wes, Ethan, Henry, Alan, James, Vincent, and Tom.  They are to be accommodated in separate houses. As host, Alper has conditions on how the guests must be placed on his planet:
 
i. His best friend Ethan must live next door to him
ii. Henry and James must not live next door to each other
iii. Alan and Vincent must live next door to each other
iv. Wes should be as many edges away from Alper as possible
v. Alper lives in the Mansion
vi. Tom lives (strictly) closer in distance to Wes than Alper
 
i. Given all conditions are met, how many combinations of living arrangements are there?
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