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Determine the number of non-negative integral solutions to \mathrm{3 x+y+z=24}.

Option: 1

215


Option: 2

117


Option: 3

 112


Option: 4

242


Answers (1)

best_answer

For \mathrm{x+y+z=24, x \geq 0, y \geq 0, z \geq 0}

Assuming \mathrm{x}=\mathrm{s}
Then \mathrm{y+z=24-3 s}
\mathrm{0 \leq 24-3 s \leq 24} and \mathrm{0 \leq s \leq 8}

The number of total integral solutions
\mathrm{={ }^{24-3 s+2-1} C_{2-1}={ }^{25-3 s} C_{1}}
\mathrm{=25-3 s}

As a result, the total number of solutions to the original problem is

\mathrm{\sum_{s=0}^{8}(25-3 s)=25 \sum_{s=0}^{8}(1)-3 \sum_{s=0}^{8} s}
\mathrm{25(9)-3\left(\frac{8 \times 9}{2}\right)=117}
Option (b) is correct.

Posted by

Pankaj Sanodiya

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