2024 AMC 12B 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} \qquad \textbf{(B) }\frac{1}{221} \qquad \textbf{(C) }\frac{1}{105} \qquad \textbf{(D) }\frac{1}{84} \qquad \textbf{(E) }\frac{1}{63}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.