Combinatorics of Street Lamp Arrangements

In summary, there are 17 street lamps along a straight street and we need to find the number of ways to turn off 5 of them without affecting regular use. We cannot turn off the end lamps or lamps adjacent to an already turned off lamp. After considering these conditions, we can calculate the number of ways by arranging 15 distinguishable lamps taken 5 at a time, and then subtracting the number of prohibited states.
  • #1
kts1230
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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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  • #2
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?
 

Related to Combinatorics of Street Lamp Arrangements

What is discrete mathematics?

Discrete mathematics is a branch of mathematics that deals with objects that can only take on distinct, separated values. It is used to study structures such as integers, graphs, and statements in logic.

What are some applications of discrete mathematics?

Discrete mathematics is used in various fields such as computer science, engineering, and cryptography. It is also used in decision-making and optimization problems in business and economics.

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Some basic concepts in discrete mathematics include sets, functions, relations, combinatorics, and graph theory. These concepts are used to study discrete structures and their properties.

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The main difference between discrete mathematics and continuous mathematics is that discrete mathematics deals with objects that have distinct, separated values, while continuous mathematics deals with objects that can take on any value within a given range.

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