A) \[\frac{2}{3}\]
B) \[\frac{3}{4}\]
C) \[\frac{47}{60}\]
D) \[\frac{49}{60}\]
Correct Answer: C
Solution :
\[=\frac{{{\left( \frac{1}{3} \right)}^{3}}+{{\left( \frac{1}{4} \right)}^{3}}+{{\left( \frac{1}{5} \right)}^{3}}-3\left( \frac{1}{3} \right)\left( \frac{1}{4} \right)\left( \frac{1}{5} \right)}{{{\left( \frac{1}{3} \right)}^{2}}+{{\left( \frac{1}{4} \right)}^{2}}+{{\left( \frac{1}{5} \right)}^{2}}-\left[ \frac{1}{3}\cdot \frac{1}{4}+\frac{1}{4}\cdot \frac{1}{5}+\frac{1}{5}\cdot \frac{1}{3} \right]}\] \[=\frac{\left( \frac{1}{3}+\frac{1}{4}+\frac{1}{5} \right)\left[ {{\left( \frac{1}{3} \right)}^{2}}+{{\left( \frac{1}{4} \right)}^{2}}+{{\left( \frac{1}{5} \right)}^{2}}-\left( \frac{1}{3}\cdot \frac{1}{4}+\frac{1}{4}\cdot \frac{1}{5}+\frac{1}{5}\cdot \frac{1}{3} \right) \right]}{{{\left( \frac{1}{3} \right)}^{2}}+{{\left( \frac{1}{4} \right)}^{2}}+{{\left( \frac{1}{5} \right)}^{2}}-\left( \frac{1}{3}\cdot \frac{1}{4}+\frac{1}{4}\cdot \frac{1}{5}+\frac{1}{5}\cdot \frac{1}{3} \right)}\]\[=\frac{1}{3}+\frac{1}{4}+\frac{1}{5}=\frac{20+15+12}{60}=\frac{47}{60}\]
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