How many ways can we turn off 5 lamps along a street?

In summary, the problem is to determine the number of ways to turn off 5 out of 17 street lamps along a straight street, while meeting certain conditions. One possible approach is to use toothpicks and a hotdog bun to represent the lamps and systematically try different combinations. Another approach is to look for a pattern in the possible combinations.
  • #1
kts1230
3
0

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 and I still don't even know where to begin. Any help would be highly appreciated.
 
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  • #2
Get out 17 toothpicks and a hotdog bun.
 
  • #3
Start with turning off #2, 4, 6, 8, & 10, then 2, 4, 6, 8, 11, then 2, 4, 6, 8, 12, etc. until you get to 2, 4, 6, 8, 16.
Then try 2, 4, 6, 9, 11, then 2, 4, 6, 9, 12, etc. to 2, 4, 6, 8, 16

Look for a pattern.
 

FAQ: How many ways can we turn off 5 lamps along a street?

1. What is discrete mathematics?

Discrete mathematics is a branch of mathematics that deals with discrete, rather than continuous, objects. It involves the study of mathematical structures that are countable or finite in nature, such as integers, graphs, and sets.

2. What are some applications of discrete mathematics?

Discrete mathematics has many practical applications in computer science, cryptography, game theory, and operations research. It is used to design algorithms, analyze networks, and solve problems in optimization and decision-making.

3. What are some key topics in discrete mathematics?

Some key topics in discrete mathematics include combinatorics, graph theory, set theory, number theory, and logic. These topics are essential for understanding and solving problems in discrete mathematics.

4. How is discrete mathematics different from other branches of mathematics?

Discrete mathematics differs from other branches of mathematics in that it deals with discrete, rather than continuous, objects. It also focuses on structures that are countable or finite in nature, whereas other branches may deal with infinite or continuous objects.

5. Are there any prerequisites for learning discrete mathematics?

Basic knowledge of algebra, geometry, and logic is helpful for understanding discrete mathematics. However, the most important prerequisite is a strong foundation in mathematical reasoning and problem-solving skills.

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