AMC 12B 2024 (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?

\text{(A)}\;\frac{1}{240}

\text{(B)}\;\frac{1}{221}

\text{(C)}\;\frac{1}{105}

\text{(D)}\;\frac{1}{84}

\text{(E)}\;\frac{1}{63}