Sure, let's simplify each of the given expressions step by step.

### a. \( \frac{1}{2} - \frac{43}{101} + \left( -\frac{1}{3} \right) - \frac{1}{6} \)

First, let's find a common denominator for all the fractions. The denominators are 2, 101, 3, and 6. The least common multiple (LCM) of these numbers is 606.

\[
\frac{1}{2} = \frac{303}{606}, \quad \frac{43}{101} = \frac{258}{606}, \quad -\frac{1}{3} = -\frac{202}{606}, \quad \frac{1}{6} = \frac{101}{606}
\]

Now, combine them:

\[
\frac{303}{606} - \frac{258}{606} - \frac{202}{606} - \frac{101}{606} = \frac{303 - 258 - 202 - 101}{606} = \frac{-258}{606} = -\frac{43}{101}
\]

So, the simplified result is:

\[
-\frac{43}{101}
\]

### b. \( \left( \frac{5}{3} - \frac{3}{7} + 9 \right) - \left( 2 + \frac{5}{7} - \frac{2}{3} \right) + \left( \frac{8}{7} - \frac{4}{3} - 10 \right) \)

First, simplify each group separately.

1. \( \frac{5}{3} - \frac{3}{7} + 9 \)

Find a common denominator for \(\frac{5}{3}\) and \(\frac{3}{7}\). The LCM of 3 and 7 is 21.

\[
\frac{5}{3} = \frac{35}{21}, \quad \frac{3}{7} = \frac{9}{21}
\]

\[
\frac{35}{21} - \frac{9}{21} = \frac{26}{21}
\]

So, \( \frac{5}{3} - \frac{3}{7} + 9 = \frac{26}{21} + 9 = \frac{26}{21} + \frac{189}{21} = \frac{215}{21} \).

2. \( 2 + \frac{5}{7} - \frac{2}{3} \)

Find a common denominator for \(\frac{5}{7}\) and \(\frac{2}{3}\). The LCM of 7 and 3 is 21.

\[
\frac{5}{7} = \frac{15}{21}, \quad \frac{2}{3} = \frac{14}{21}
\]

\[
2 + \frac{15}{21} - \frac{14}{21} = 2 + \frac{1}{21} = \frac{42}{21} + \frac{1}{21} = \frac{43}{21}
\]

3. \( \frac{8}{7} - \frac{4}{3} - 10 \)

Find a common denominator for \(\frac{8}{7}\) and \(\frac{4}{3}\). The LCM of 7 and 3 is 21.

\[
\frac{8}{7} = \frac{24}{21}, \quad \frac{4}{3} = \frac{28}{21}
\]

\[
\frac{24}{21} - \frac{28}{21} = -\frac{4}{21}
\]

So, \( \frac{8}{7} - \frac{4}{3} - 10 = -\frac{4}{21} - 10 = -\frac{4}{21} - \frac{210}{21} = -\frac{214}{21} \).

Now, combine all three groups:

\[
\frac{215}{21} - \frac{43}{21} + \left( -\frac{214}{21} \right) = \frac{215 - 43 - 214}{21} = \frac{-42}{21} = -2
\]

So, the simplified result is:

\[
-2
\]

### c. \( A = \frac{1}{2} + \frac{5}{6} + \frac{1}{3} \)

Find a common denominator for \(\frac{1}{2}\), \(\frac{5}{6}\), and \(\frac{1}{3}\). The LCM of 2, 6, and 3 is 6.

\[
\frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6}
\]

\[
\frac{3}{6} + \frac{5}{6} + \frac{2}{6} = \frac{10}{6} = \frac{5}{3}
\]

So, the simplified result is:

\[
A = \frac{5}{3}
\]

### d. \( B = \frac{1}{2} - \left[ \frac{3}{8} + \left( -\frac{7}{4} \right) \right] \)

First, simplify the expression inside the brackets:

\[
\frac{3}{8} + \left( -\frac{7}{4} \right) = \frac{3}{8} - \frac{14}{8} = -\frac{11}{8}
\]

Now, subtract this from \(\frac{1}{2}\):

\[
\frac{1}{2} - \left( -\frac{11}{8} \right) = \frac{1}{2} + \frac{11}{8}
\]

Find a common denominator for \(\frac{1}{2}\) and \(\frac{11}{8}\). The LCM of 2 and 8 is 8.

\[
\frac{1}{2} = \frac{4}{8}
\]

\[
\frac{4}{8} + \frac{11}{8} = \frac{15}{8}
\]

So, the simplified result is:

\[
B = \frac{15}{8}
\]

### e. \( \frac{2}{3} + -\frac{10}{6} + -\frac{4}{3} \)

First, simplify \(-\frac{10}{6}\):

\[
-\frac{10}{6} = -\frac{5}{3}
\]

Now, combine all the fractions:

\[
\frac{2}{3} - \frac{5}{3} - \frac{4}{3} = \frac{2 - 5 - 4}{3} = \frac{-7}{3}
\]

So, the simplified result is:

\[
-\frac{7}{3}
\]

### f. \( \frac{7}{3} - \frac{5}{6} - \frac{2}{3} \)

Find a common denominator for \(\frac{7}{3}\), \(\frac{5}{6}\), and \(\frac{2}{3}\). The LCM of 3 and 6 is 6.

\[
\frac{7}{3} = \frac{14}{6}, \quad \frac{2}{3} = \frac{4}{6}
\]

Now, combine all the fractions:

\[
\frac{14}{6} - \frac{5}{6} - \frac{4}{6} = \frac{14 - 5 - 4}{6} = \frac{5}{6}
\]

So, the simplified result is:

\[
\frac{5}{6}
\]

These are the simplified results for each expression.