Q4 = probability that 4 depth charges miss the submarine
$Q_4 = \left( \frac{1}{4} \right)^4$
$Q_4 = \frac{1}{256}$
Q3 = probability that 3 depth charges miss the submarine and 1 damage
$Q_3 = \left( \frac{1}{4} \right)^3 \cdot \left( \frac{1}{4} \right)^1 \times {^4}C_3$
$Q_3 = \frac{1}{64}$
Qescape = Probability that the submarine will escape in 4 depth charges
$Q_\text{escape} = Q_3 + Q_4$
$Q_\text{escape} = \frac{5}{256}$
Psunk = Probability that 4 depth charges will sink the submarine
$P_\text{sunk} = 1 - Q_\text{escape}$
$P_\text{sunk} = \frac{251}{256}$ ← answer