Question 1160018
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In other form, the question is


    How many solutions does this equation have

        {{{x[1]}}} + {{{x[2]}}} + {{{x[3]}}} = 9

    in positive integer numbers {{{x[1]}}}, {{{x[2]}}} and {{{x[3]}}} ?


The solution can be done using the "Stars and Bars method", which gives in this case the answer  

         {{{C[9-1]^(3-1)}}} = {{{C[8]^2}}} = {{{(8*7)/2}}} = 4*7 = 28.


<U>ANSWER</U>.  28 different solutions   //  (accounting for all possible permutations, cases and sub-cases . . . )
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Solved and answered.


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For the "Stars and Bars method", see the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/misc/Stars-and-bars-method-for-Combinatorics-problems.lesson>Stars and bars method for Combinatorics problems</A> 

in this site, 


or, alternatively, this Wikipedia article


<A HREF=https://en.wikipedia.org/wiki/Stars_and_bars_%28combinatorics%29>https://en.wikipedia.org/wiki/Stars_and_bars_%28combinatorics%29</A>


https://en.wikipedia.org/wiki/Stars_and_bars_%28combinatorics%29