Probability

Algebra Level 5

Let $$P(x) = x^2 - 3x - 9$$. A real number $$x$$ is chosen at random from the interval $$5 \le x \le 15$$. The probability that $$\lfloor\sqrt{P(x)}\rfloor = \sqrt{P(\lfloor x \rfloor)}$$ is equal to $$\frac{\sqrt{a} + \sqrt{b} + \sqrt{c} - d}{e}$$, where $$a, b, c, d$$, and $$e$$ are positive integers. Find $$a + b + c + d + e$$.

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