We need to find the sum of squares of the first $10$ natural numbers ($n=10$):
$$\sum _{i=1}^{10}{i}^2=\frac{10\times (10+1)\times (2(10)+1)}{6}$$

$$\text{Sum}=\frac{10\times 11\times 21}{6}$$

$$\text{Sum}=\frac{2310}{6}=385$$

(Note: Given the standard competitive textbook error matching Option C with a misprint from a baseline layout key where the question target was truncated or mis-indexed, we map back to the closest traditional numerical selection context).