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  • Determine how many non-negative integral solutions there are for <img alt="\mathrm{6 x+y+z=30}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B6%20x&plus;y&plus;z%3D30%7D"

Determine how many non-negative integral solutions there are for \mathrm{6 x+y+z=30}.

Option: 1

450


Option: 2

200


Option: 3

100


Option: 4

180


Answers (1)

best_answer

For \mathrm{6 x+y+z=30, x \geq 0, y \geq 0, z \geq 0}

Assuming \mathrm{x}=\mathrm{s}
Then \mathrm{y+z=30-6 s}

\mathrm{0 \leq 30-6 s \leq 30} and \mathrm{0 \leq s \leq 5}

The number of total integral solutions
\mathrm{={ }^{30-6 s+2-1} C_{2-1} }
\mathrm{={ }^{31-6 s} C_{1} }
\mathrm{=31-6 s }

As a result, the total number of solutions to the original problem is
\mathrm{=\sum_{n=0}^{5}(31-6 s) }
\mathrm{=31 \sum_{n=0}^{5}(1)-6 \sum_{n=0}^{5} s}
\mathrm{=30(9)-6\left(\frac{5 \times 6}{2}\right)}
\mathrm{=270-90}
\mathrm{=180}

Option (d) is correct.

Posted by

Sanket Gandhi

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