The value of:  \(\frac{5}{7}\ of\ \left[\frac{1}{1\frac{3}{7}}+\frac{6}{5}\ of\ \frac{3\frac{1}{3}-2\frac{1}{2}}{2\frac{5}{21}-2}\right]+\left[\frac{7}{5-2\frac{2}{3}}\div \frac{3-\frac{2}{3-1\frac{1}{2}}}{4-1\frac{1}{2}}\right]\)

Given:

\(\frac{5}{7}\ of\ \left[\frac{1}{1\frac{3}{7}}+\frac{6}{5}\ of\ \frac{3\frac{1}{3}-2\frac{1}{2}}{2\frac{5}{21}-2}\right]+\left[\frac{7}{5-2\frac{2}{3}}\div \frac{3-\frac{2}{3-1\frac{1}{2}}}{4-1\frac{1}{2}}\right]\)

Calculations:

\(\frac{5}{7}\ of\ \left[\frac{1}{1\frac{3}{7}}+\frac{6}{5}\ of\ \frac{3\frac{1}{3}-2\frac{1}{2}}{2\frac{5}{21}-2}\right]+\left[\frac{7}{5-2\frac{2}{3}}\div \frac{3-\frac{2}{3-1\frac{1}{2}}}{4-1\frac{1}{2}}\right]\)

⇒ \(\frac{5}{7} \times \left[\frac{1}{\frac{10}{7}}+\frac{6}{5}\times \frac{\frac{10}{3}-\frac{5}{2}}{\frac{47}{21}-2}\right]+\left[\frac{7}{5-\frac{8}{3}}\div \frac{3-\frac{2}{3-\frac{3}{2}}}{4-\frac{3}{2}}\right]\)

⇒ \(\frac{5}{7} \times \left[\frac{7}{10}+\frac{6}{5}\times \frac{\frac{20}{6}-\frac{15}{6}}{\frac{47 - 42}{21}}\right]+\left[\frac{7}{\frac{15 - 8}{3}}\div \frac{3-\frac{2}{\frac{6 -3}{2}}}{\frac{ 8 - 3}{2}}\right]\)

⇒ \(\frac{5}{7} \times \left[\frac{7}{10}+\frac{6}{5}\times \frac{\frac{5}{6}}{\frac{5}{21}}\right]+\left[\frac{7}{\frac{7}{3}}\div \frac{3-\frac{4}{3}}{\frac{5}{2}}\right]\)

⇒ \(\frac{5}{7} \times \left[\frac{7}{10}+\frac{6}{5}\times \frac{21}{6}\right]+\left[3\div \frac{\frac{5}{3}}{\frac{5}{2}}\right]\)

⇒ \(\frac{5}{7} \times \left[\frac{7}{10}+\frac{21}{5}\right]+\left[3\div \frac{2}{3}\right]\)

⇒ \(\frac{5}{7} \times \left[\frac{7}{10}+\frac{42}{10}\right]+\left[3\times \frac{3}{2}\right]\)

⇒ \(\frac{5}{7} \times \left[\frac{49}{10}\right]+\left[\frac{9}{2}\right]\)

⇒ \(\frac{5}{7} \times \left[\frac{49}{10}\right]+\left[\frac{9}{2}\right]\)

⇒ \(\left[\frac{7}{2}\right]+\left[\frac{9}{2}\right]\)

⇒ \(\frac{16}{2}\)

⇒ 8

∴ The correct answer is option 3.