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What is the total number of positive integer solutions to the equation $(x_{1}+x_2+x_3)$$(y_1+y_2+y_3+y_4)$ = 15 ? Will I have to decompose the equation into equation of $x$ and equation of $y$ then solve ? Then I also have a problem in decomposing the right hand side. How to solve such type of problems? Any insight? Thank you in advance.

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    $\begingroup$ Just to add something to Paolo's answer, you'll be better off if you know 'Stars And Bars' theorem for solving general versions of the above problem. - en.wikipedia.org/wiki/Stars_and_bars_(combinatorics) $\endgroup$
    – Naive
    Jun 6, 2017 at 11:35
  • $\begingroup$ That's helpful.Thanks a lot. $\endgroup$ Jun 6, 2017 at 11:39

1 Answer 1

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Note that $\sum x_i \ge 3$ and $\sum y_i \ge 4$. Considering the factorizations of $15$, we have $$ 15=\underbrace{(x_1+x_2+x_3)}_{=3}\underbrace{(y_1+y_2+y_3+y_4)}_{=5}. $$ Hence all the variables are $1$, except one $y_i$. This can be done in $4$ ways.

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  • $\begingroup$ Thanks,nice explanation. $\endgroup$ Jun 6, 2017 at 10:45
  • $\begingroup$ $$=1;=15$$ $$=-1;=-15$$ $$=3;=5$$ $$=-3;=-5$$ $$=5;=3$$ $$=-5;=-3$$ $$=15;=1$$ $$=-15;=-1$$ $\endgroup$
    – individ
    Jun 6, 2017 at 10:48
  • $\begingroup$ Sorry ,what is this? ,I can't get it .. $\endgroup$ Jun 6, 2017 at 11:27
  • $\begingroup$ @individ the variables $x_i$ are $y_i$ are positive.. $\endgroup$ Jun 6, 2017 at 12:01

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