Let's simplify the given expression step by step:
The expression is:
\[
(16−12)^2 + \frac{5(8+2^2)}{5^3} − (1+4)^2
\]
1. Calculate \( (16 - 12)^2 \):
\[
(16 - 12) = 4 \implies (4)^2 = 16
\]
2. Calculate \( 2^2 \):
\[
2^2 = 4
\]
3. Calculate \( 8 + 2^2 \):
\[
8 + 4 = 12
\]
4. Calculate \( 5(8 + 2^2) \):
\[
5 \cdot 12 = 60
\]
5. Calculate \( 5^3 \):
\[
5^3 = 125
\]
6. Now compute \( \frac{5(8 + 2^2)}{5^3} \):
\[
\frac{60}{125} = \frac{12}{25}
\]
7. Calculate \( (1 + 4)^2 \):
\[
(1 + 4) = 5 \implies (5)^2 = 25
\]
Now combine all the parts calculated:
\[
16 + \frac{12}{25} - 25
\]
Convert \( 16 \) and \( 25 \) to have a common denominator of \( 25 \):
\[
16 = \frac{400}{25}, \quad 25 = \frac{625}{25}
\]
So now we have:
\[
\frac{400}{25} + \frac{12}{25} - \frac{625}{25}
\]
Combine the fractions:
\[
\frac{400 + 12 - 625}{25} = \frac{-213}{25}
\]
The numerator is \(-213\).
None of the responses provided match this, indicating a possible error in interpretation or calculations. If we only need the simplified numerator without considering the entire expression at this moment, it would have been \(-213\).
Since there is no option that represents this, please double-check any context given for the numerator, or the expression itself.
However, if you meant to evaluate the expression differently or needed further clarification, please let me know!