To simplify the expression \(\frac{1}{2} + \left\{4\frac{3}{4} - \left(3\frac{1}{6} - 2\frac{1}{3}\right)\right\}\), follow these steps:

1. Calculate \(3\frac{1}{6} - 2\frac{1}{3}\):

First, convert the mixed numbers to improper fractions:
\[
3\frac{1}{6} = \frac{19}{6}
\]
\[
2\frac{1}{3} = \frac{7}{3}
\]

Find a common denominator for \(\frac{19}{6}\) and \(\frac{7}{3}\). The least common denominator is 6:
\[
\frac{7}{3} = \frac{14}{6}
\]

Now subtract:
\[
\frac{19}{6} - \frac{14}{6} = \frac{5}{6}
\]

2. Subtract the result from \(4\frac{3}{4}\):

Convert \(4\frac{3}{4}\) to an improper fraction:
\[
4\frac{3}{4} = \frac{19}{4}
\]

Convert \(\frac{5}{6}\) to a fraction with denominator 4:
\[
\frac{5}{6} = \frac{10}{12} \text{ (Equivalent fraction)}
\]
\[
\frac{5}{6} = \frac{5}{6} = \frac{10}{12} = \frac{10}{12} = \frac{10}{12} = \frac{5}{6} = \frac{5}{6} = \frac{10}{12}
\]

Now subtract:
\[
\frac{19}{4} - \frac{5}{6}
\]
Find a common denominator, which is 12:
\[
\frac{19}{4} = \frac{57}{12}
\]
\[
\frac{5}{6} = \frac{10}{12}
\]
Subtract:
\[
\frac{57}{12} - \frac{10}{12} = \frac{47}{12}
\]

3. Add \(\frac{1}{2}\) to \(\frac{47}{12}\):

Convert \(\frac{1}{2}\) to a fraction with denominator 12:
\[
\frac{1}{2} = \frac{6}{12}
\]

Add:
\[
\frac{47}{12} + \frac{6}{12} = \frac{53}{12}
\]

Convert \(\frac{53}{12}\) to a mixed number:
\[
\frac{53}{12} = 4\frac{5}{12}
\]


The final result is \(4 \frac{5}{12}\).

The correct answer is C. 4 \frac{5}{12}.