\( \Large \frac{\frac{1}{3}.\frac{1}{3}.\frac{1}{3}+\frac{1}{4}.\frac{1}{4}.\frac{1}{4}-3\frac{1}{3}.\frac{1}{4}.\frac{1}{5}+\frac{1}{5}.\frac{1}{5}.\frac{1}{5}}{\frac{1}{3}.\frac{1}{3}+\frac{1}{4}.\frac{1}{4}+\frac{1}{5}.\frac{1}{5}-\left(\frac{1}{3}.\frac{1}{4}+\frac{1}{4}.\frac{1}{5}+\frac{1}{5}.\frac{1}{3}\right) } \) is
Correct Answer: |
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C) \( \Large \frac{47}{60} \) |
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Description for Correct answer:
\( \Large \frac{\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3}+\frac{1}{4} \times \frac{1}{4} \times \frac{1}{4}-3 \times \frac{1}{3} \times \frac{1}{4} \times \frac{1}{5}+\frac{1}{5} \times \frac{1}{5} \times \frac{1}{5}}{\frac{1}{3} \times \frac{1}{3}+\frac{1}{4} \times \frac{1}{4}+\frac{1}{5} \times \frac{1}{5}- \left(\frac{1}{3} \times \frac{1}{4}+\frac{1}{4} \times \frac{1}{5}+\frac{1}{5} \times \frac{1}{3}\right) } \)
\( \Large A^{3}+B^{3}+C^{3}-3ABC = \left(A+B+C\right) \) \( \Large \left(A^{2}+B^{2}+c^{2}-AB-BC-CA\right) \)
Therefore, \( \Large \frac{ \left(\frac{1}{3}\right)^{3}+ \left(\frac{1}{4}\right)^{3}-3 \times \frac{1}{3} \times \frac{1}{4} \times \frac{1}{5}+ \left(\frac{1}{5}\right)^{3} }{ \left(\frac{1}{3}\right)^{2}+ \left(\frac{1}{4}\right)^{2}+ \left(\frac{1}{5}\right)^{2}-\frac{1}{3} \times \frac{1}{4}-\frac{1}{4} \times \frac{1}{5}-\frac{1}{5} \times \frac{1}{3} } \)
=\( \Large \frac{ \left(\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\right)\left[ \left(\frac{1}{3}\right)^{2}+ \left(\frac{1}{4}\right)^{2}+ \left(\frac{1}{5}\right)^{2}-\frac{1}{3} \times \frac{1}{4}-\frac{1}{4} \times \frac{1}{5}-\frac{1}{5} \times \frac{1}{3} \right] }{\left[ \left(\frac{1}{3}\right)^{2}+ \left(\frac{1}{4}\right)^{2}+ \left(\frac{1}{5}\right)^{2}-\frac{1}{3} \times \frac{1}{4}-\frac{1}{4} \times \frac{1}{5}-\frac{1}{5} \times \frac{1}{3} \right]} \)
= \( \Large \frac{20+15+12}{60} = \frac{47}{60} \)
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