Evaluate the given mathematical expression:

$14\frac{4}{5}+6\frac{3}{10}-8\frac{7}{10}-3\frac{2}{5}=?$

The correct option is 9.

To evaluate the given mathematical expression involving mixed fractions, we can group the whole numbers (integer parts) and the fractional parts separately.

The given expression is:

$14\frac{4}{5}+6\frac{3}{10}-8\frac{7}{10}-3\frac{2}{5}$

Step 1: Group the whole numbers and fractions together.
Separate the integer values and the fractions into two separate groups:

$(14+6-8-3)+(\frac{4}{5}+\frac{3}{10}-\frac{7}{10}-\frac{2}{5})$

Step 2: Simplify the whole number part.

$14+6-8-3=20-11=9$

Step 3: Simplify the fractional part.
Find a common denominator for all fractions. The denominators are 5 and 10, so the least common denominator is 10.

Convert fractions with denominator 5 into equivalent fractions with denominator 10:

$\frac{4}{5}=\frac{4\times 2}{5\times 2}=\frac{8}{10}$

$\frac{2}{5}=\frac{2\times 2}{5\times 2}=\frac{4}{10}$

Now substitute these fractions back into the fractional expression:

$\frac{8}{10}+\frac{3}{10}-\frac{7}{10}-\frac{4}{10}$

Combine the numerators over the common denominator 10:

$\frac{8+3-7-4}{10}=\frac{11-11}{10}=\frac{0}{10}=0$

Step 4: Combine the integer and fractional results.

$9+0=9$

Therefore, the value of the given expression is 9.