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I'm having a slight problem with a simple equation of the sort $a_1+a_2+a_3...=n$. Where $n,a_1, a_2, a_3... \in N$. I do know how to find the number of solutions to these equations when they are of the regular form. But if you apply a constraint, I'm unable to solve it.

$a_1+a_2+a_3+a_4+a_5+a_6=14$

The above equation has the constraints that all of them are whole numbers and that none of them are greater than 4. I need to find the number of solutions to the problem.

Could I please get some help regarding this problem?

Any help would be appreciated. Thank you.

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HINT: Since you can solve the regular form substitute $a_i = b_i -1$ where $b_i \in \mathbb N$. It is in the 'regular' form now. What similar thing can you do to deal with the $a_i \le 4$ constraint?

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  • $\begingroup$ Hmmm Nope, I don't get quite what you mean. I understand how to apply the constraint that $a_1$ should always be a positive integer ($a_1$ >= 1). But not that it should be lesser than some value. $\endgroup$
    – nobody_nowhere
    Jan 11, 2015 at 20:18
  • $\begingroup$ Substituting $a_i = b_i - 1$ would just complicate my problem, it'll change my equation to $b_1 + b_2 + b_3 + b_4 + b_5 + b_6 = 20$ but now, $b_i$ is $\leq$ 5 now. :/ I simply don't get it :/ $\endgroup$
    – nobody_nowhere
    Jan 11, 2015 at 20:23

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