How are these two equal?(equation, inequality)

  • Thread starter Pithikos
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  • #1
Pithikos
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How are these two equal??(equation, inequality)

I study discrete mathematics and we are doing combinations at the moment. There is this example in the book(Discrete Mathematics and Combinatorics p. 30 Ex. 1.43) where it states that the number of integer solutions for:

x1+x2+x3+x4+x5+x6<10 where xi[tex]\geq[/tex]0

is equal to the number of integer solutions of

x1+x2+x3+x4+x5+x6+x7=10 where xi[tex]\geq[/tex]0 and x7>0

Can someone explain me this? The author supposes that I magically understand what goes through his mind.

Here is a screenshot of the problem: http://img251.imageshack.us/img251/823/garbageab.jpg [Broken]
 
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Answers and Replies

  • #2
saim_
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This is subtle, but, trivial: Try to see that to all the possbile solutions of x1+x2+x3+x4+x5+x6<10, we can add a positive number and make it equal to 10 and also that it is only the set of these solutions to which we can add a positive number and make it equal to 10; thus the set of integer solutions of both equations have the same number of elements.
 

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