Combinatorics of Street Lamp Arrangements

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SUMMARY

The problem involves arranging 17 street lamps with specific constraints on which can be turned off. Only 5 lamps can be turned off, with the stipulation that the end lamps must remain on and no two adjacent lamps can be turned off. The solution begins by focusing on the 15 lamps that can be manipulated, leading to a combinatorial analysis that accounts for the restrictions imposed by the adjacency rule.

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  • Understanding of combinatorial mathematics
  • Familiarity with distinguishable arrangements
  • Knowledge of constraints in combinatorial problems
  • Basic principles of counting and permutations
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  • Study combinatorial techniques for counting arrangements with restrictions
  • Learn about the principle of inclusion-exclusion in combinatorics
  • Explore the concept of generating functions for combinatorial problems
  • Investigate similar problems involving arrangements with adjacency constraints
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Homework Statement



There are 17 street lamps along a straight street. In order to save electricity and not affect the regular use at the same time, we can shut down 5 of these lamps. But we cannot turn off a lamp at either end of the street, and we cannot turn off a lamp adjacent to a lamp that is already off. Under such conditions, in how many ways can we turn off 5 lamps?

Homework Equations





The Attempt at a Solution



I've looked at this question a few times now and I still don't even know where to begin. Any help would be highly appreciated.
 
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Hmm .. I would start as follows:

First, forget about the two end-lights. They're always on. That leaves 15 lights to worry about.

How many ways can you arrange 15 distinguishable things taken 5 at a time?

Then, how many states are prohibited by the "no 2 adjacent lights off" rule?
 

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