Number of integer solutions such that each number is less than 7

Asked Nov 12 '20 at 07:35

Active Nov 12 '20 at 09:18

Viewed 32 times

Find the integer solutions to $$x_1 + \dots x_{10} = 25$$ such that $$1 \leq x_i \leq 6$$

I made transformation $$y_i = 7 - x_i$$ and got the new problem as $$y_1 + \dots + y_{10} = 45$$ but in this case I can have $$y_i = 8$$ which would mean that $$x_i = -1$$. How do I solve this then?

Edit: any other solution is appreciated.

edited Nov 12 '20 at 09:18

N. F. Taussig

76,571

asked Nov 12 '20 at 07:35

user601297

1,106

Well, one way of obtaining the number of solutions, would be to calculate the coefficient of $x^{25}$ in $$ (x^1+x^2 + x^3 + x^4 + x^5+ x^6)^{10} $$ – Matti P. Nov 12 '20 at 08:04
I don’t understand how this solution works. – user601297 Nov 12 '20 at 08:06
Do you need to find the solutions or the number of solutions? To me, it looks like a pretty standard combinatorics question that asks to find the number of solutions. – Matti P. Nov 12 '20 at 08:12
@MattiP. Number of solutions – user601297 Nov 12 '20 at 08:13
Notice also this similar question: https://math.stackexchange.com/questions/203835/enumerating-number-of-solutions-to-an-equation – Matti P. Nov 12 '20 at 08:14
Did you mean nonnegative integer solutions or positive integer solutions? In either case, you can apply the Inclusion-Exclusion Principle. First, find the number of solutions without restriction, then eliminate those cases in which or more of the variables assumes a value greater than $6$. – N. F. Taussig Nov 12 '20 at 09:21
@N.F.Taussig yes i think I’ve got it now, thanks a lot. – user601297 Nov 12 '20 at 09:21

Number of integer solutions such that each number is less than 7

0 Answers0