Question 1199982
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<pre>

It is the same as to ask


    +-------------------------------------------------------------+
    |  how many solutions in integer non-negative numbers         |
    |                                                             |
    |  this equation does have  {{{x[1]}}} + {{{x[2]}}} + {{{x[3]}}} = 10.            |
    +-------------------------------------------------------------+


The "stars and bars" method gives the answer: the number of solutions for this equation is

    {{{C[n+k-1]^n}}} = {{{C[10+3-1]^10}}} = {{{C[12]^10}}} = {{{C[12]^2}}} = {{{(12*11)/(1*2)}}} = 6*11 = 66.    <U>ANSWER</U>


In this formula, n= 10, k = 3.
</pre>

Solved.


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For the stars and bars method see this Wikipedia article

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



See also the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =https://www.algebra.com/algebra/homework/Permutations/Stars-and-bars-method-for-Combinatorics-problems-2.lesson>Stars and bars method for Combinatorics problems</A> 

in this site.