Question #41026

Let $f(x) = \begin{cases} x^3 - 3x + k - 2 & \text{for } -2 \le x < 1 \\ ax^2 + bx + c & \text{for } 1 \le x \le 3 \end{cases}$ be a function continuous and differentiable on $(-2, 3)$, where $a, b, c$ are real constants and $f(3) = k$. Let $S$ be the set of all integers $k$ such that the equation $f(x) = 0$ has exactly 3 distinct real roots. For each $k \in S$, a bag $B_k$ is prepared containing $N_k$ red balls and $15 - N_k$ black balls, where $N_k$ is the number of non-negative integer solutions to the equation $x + y + z = k$. A bag is selected at random (all bags in $S$ are equally likely to be chosen), and two balls are drawn from it without replacement. If both balls are red, let $P$ be the probability that they were drawn from bag $B_3$. Find the value of $P$.