1/3.1/3.1/3 + 1/4.1/4.1/4 - 3.1/3.1/4.1/5 + 1/5.1/5.1/51/3.1/3 + 1/4.1/4 + 1/5.1/5 - ( 1/3.1/4 + 1/4.1/5 + 1/5.1/3 ) is?

Answer: Option C

$$\frac{\frac{1}{3}.\frac{1}{3}.\frac{1}{3}+\frac{1}{4}.\frac{1}{4}.\frac{1}{4}-3.\frac{1}{3}.\frac{1}{4}.\frac{1}{5}+\frac{1}{5}.\frac{1}{5}.\frac{1}{5}}{\frac{1}{3}.\frac{1}{3}+\frac{1}{4}.\frac{1}{4}+\frac{1}{5}.\frac{1}{5}-\left(\frac{1}{3}.\frac{1}{4}+\frac{1}{4}.\frac{1}{5}+\frac{1}{5}.\frac{1}{3}\right)}$$

A³ + B³ + C³ - 3ABC = (A + B + C)(A² + B² + C² - AB - BC - CA)

$$\therefore \frac{{\left(\frac{1}{3}\right)}^3+{\left(\frac{1}{4}\right)}^3-3.\frac{1}{3}.\frac{1}{4}.\frac{1}{5}+{\left(\frac{1}{5}\right)}^3}{{\left(\frac{1}{3}\right)}^2+{\left(\frac{1}{4}\right)}^2+{\left(\frac{1}{5}\right)}^2-\frac{1}{3}.\frac{1}{4}-\frac{1}{4}.\frac{1}{5}-\frac{1}{5}.\frac{1}{3}}$$
$$=\frac{\left(\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\right)\left[{\left(\frac{1}{3}\right)}^2+{\left(\frac{1}{4}\right)}^2+{\left(\frac{1}{5}\right)}^2-\frac{1}{3}.\frac{1}{4}-\frac{1}{4}.\frac{1}{5}-\frac{1}{5}.\frac{1}{3}\right]}{\left[{\left(\frac{1}{3}\right)}^2+{\left(\frac{1}{4}\right)}^2+{\left(\frac{1}{5}\right)}^2-\frac{1}{3}.\frac{1}{4}-\frac{1}{4}.\frac{1}{5}-\frac{1}{5}.\frac{1}{3}\right]}$$
$$\begin{array}{rl}& =\frac{20+15+12}{60}\\ & =\frac{47}{60}\end{array}$$