Factorial Fire

Algebra Level 3

\[\displaystyle \frac{1}{2! 17!} + \frac{1}{3! 16!} + \frac{1}{4! 15!} + \frac{1}{5! 14!} + \frac{1}{6! 13!} + \frac{1}{7! 12!} + \frac{1}{8! 11!} + \frac{1}{9! 10!} = \frac{2^a - b}{c!}\]

If the above equation holds true for positive integers \(a\), \(b\), and \(c\), find \(a + b + c\).

Notation: \(!\) denotes the factorial notation. For example, \(3! = 3 \times 2 \times 1\).

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