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When a die is rolled, one of the first six positive integers is obtained. Suppose that the die is rolled five times and the sum of the five integers thus obtained is added. The five throws constitute a trial. Find the number of possible trials such that the sum is at most $12$.

This can be written as $x_1+x_2+x_3+x_4+x_5\leq 12$ where $1\leq x_i\leq6$ for $i=1,2,...,5$. However, this restriction is making this problem difficult for me. I know that I can figure out the solution by considering the generating function $G(x)=(x+x^2+x^3+x^4+x^5+x^6)^5$ and take the sum of all the coefficients that are less than or equal to $12$, but I want to know how to do this with the stars and bars technique.

I know that I can rewrite the linear inequality as $x_1'+x_2'+x_3'+x_4'+x_5'\leq 7$ so we can count the number of positive integral solutions, but then we have still over counted and we need to use the restriction $x_i\leq 6$. I do not know how to do this. Any solutions or hints are greatly appreciated.

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  • First, to handle the $\leq 7$ aspect of it, introduce an additional variable $x_6 = 7-x_1'-x_2'-x_3'-x_4'-x_5'$ and note that $x_1'+x_2'+\dots+x_5'+x_6 = 7$ and that $0\leq x_6$. Now you can deal with equality instead of an inequality. Next, to handle the $x_i\leq 6$, or more correctly now $x_i'\leq 5$, use inclusion-exclusion. Find how many of the solutions were "bad" because one or more of the $x_i'$ were too big and remove those from your count. – JMoravitz Apr 07 '21 at 12:06
  • If $x_1'$ were too big, then it would have been false that $0\leq x_1'\leq 5$ meaning it would have been true that $x_1'\geq 6$. You can use a similar technique to shift the total to count how many cases there were for this to happen. In a larger problem, you may have to continue with inclusion-exclusion finding those which were "bad" because two or more upper bounds were violated. See your textbook for more information on what to do in such a case. – JMoravitz Apr 07 '21 at 12:07
  • In the equation involving $\le7$ you probably meant non-negative solutions with the restriction $x_i\le5$. – user Apr 07 '21 at 12:18

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