AMC 12 2024b Problem 17

Integers $a$ and $b$ are randomly chosen without replacement from the set of integers with absolute value not exceeding $10$. What is the probability that the polynomial $x^3 + ax^2 + bx + 6$ has $3$ distinct integer roots?

$\textbf{(A) }\frac{1}{240} \textbf{(B) }\frac{1}{221} \textbf{(C) }\frac{1}{105} \textbf{(D) }\frac{1}{84} \textbf{(E) }\frac{1}{63}$