The equation is: $$x_1+x_2+x_3+x_4+x_5+x_6<10$$ with $x_i\geq 0$ for $i=1,2,\dots,6$.
- What is the number of integer solutions of the following equation ?
- Find the solution without computing 9 combinations.
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Robert Z
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Merve Gürsu
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2 Answers
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The equation is equivalent to $$x_1+x_2+x_3+x_4+x_5+x_6+y=9$$ with $x_i\geq 0$ and $y\geq 0$. Then, by using stars-and-bars, we get $$\binom{9+6}{6}.$$
Robert Z
- 145,942
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@Merve Gürsu Any further doubt? – Robert Z Dec 07 '16 at 17:13
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I think the answer is 5005. Is that all? – Merve Gürsu Dec 07 '16 at 19:31
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Okay, sorry :) Thank you for all your assistance. – Merve Gürsu Dec 07 '16 at 19:38
0
Write
$$x_1+x_2+...+x_6=9-y$$
with $y \ge 0$ and solve
$$x_1+x_2+...+x_6+y=9$$
for integer solution.
Arnaldo
- 21,342