Combinatorics Question

In summary, the sum of 5 positive real numbers is 100. Two numbers among them have a difference of at most 10, so the sum is at least 102.
  • #1
rbzima
84
0

Homework Statement



The sum of 5 positive real numbers is 100. Prove that there are two numbers among them whose difference is at most 10.

Homework Equations



Nothing really...

The Attempt at a Solution



The biggest problem I'm running into is that I can think of specific examples, but translating that into an algebraic argument has always been my weak area. Getting started is where I struggle the most... but I'm thinking the following:

Let's assume there are no two positive real numbers whose difference is at most 10.
Let [tex]a_{1}, a_{2}, a_{3}, a_{4}, a_{5}[/tex] each represent some positive real number whose sum equal 100. Given that [tex]a_{1}[/tex] is the smallest real number, [tex]a_{2} = 10 + a_{1}[/tex], [tex]a_{3} = 10 + a_{2}[/tex] and so on...

Don't really know where to go from here! Suggestions?
 
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  • #2
Well you've got a finite set of numbers, so you know it has a minimum. Let [itex]x_1[/itex] be that minimum. Furthermore they're real numbers so they're ordered. Let [itex]x_1\leq x_2 \leq x_3 \leq x_4 \leq x_5[/itex]. Now suppose that there do not exist a pair of numbers such that their difference is at most 10.

Try to show that [itex]x_1+x_2+x_3+x_4+x_5>100[/itex].
 
  • #3
Tom Mattson said:
Well you've got a finite set of numbers, so you know it has a minimum. Let [itex]x_1[/itex] be that minimum. Furthermore they're real numbers so they're ordered. Let [itex]x_1\leq x_2 \leq x_3 \leq x_4 \leq x_5[/itex]. Now suppose that there do not exist a pair of numbers such that their difference is at most 10.

Try to show that [itex]x_1+x_2+x_3+x_4+x_5>100[/itex].

Wow, that was straight forward... Totally figured it out...
I had an epiphany right before you responded, so I should be good now. I was definitely thinking the same thing, and ended up solving in terms of [tex]a_{1}[/tex]. Regardless of how small [tex]a_{1}[/tex] is, as long as it's not 0, you will receive some value that is greater than 100, thus we have a contradiction, which means that there must be at least two values who have a difference of at most 10!

Thanks for the help!
 

1) What is combinatorics?

Combinatorics is a branch of mathematics that deals with counting, arrangements, and combinations of objects or events.

2) What are some real-world applications of combinatorics?

Combinatorics has numerous applications in fields such as computer science, biology, economics, and physics. Some examples include analyzing the efficiency of algorithms, designing DNA sequences for genetic engineering, predicting stock market trends, and studying the behavior of particles in quantum mechanics.

3) How is combinatorics different from probability theory?

Combinatorics focuses on counting and arranging objects or events without considering their likelihood of occurrence, while probability theory deals with the likelihood or chance of events occurring. However, combinatorics and probability theory are closely related and often used in conjunction with each other.

4) What are some common techniques used in combinatorics?

Some commonly used techniques in combinatorics include the multiplication principle, combinations and permutations, the inclusion-exclusion principle, and generating functions.

5) How can I improve my problem-solving skills in combinatorics?

The best way to improve problem-solving skills in combinatorics is through practice. Start with simple problems and gradually work your way up to more complex ones. It is also helpful to familiarize yourself with common techniques and strategies used in combinatorics, and to study and solve problems from different sources.

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