Combinatorics Fever

Consider a game played by \(10\) people in which each flips a fair coin at the same time. If all but one of the coin comes up at the same, then odd person wins (e.g. if there are nine tails and one head then head wins). If such a situation does not occur, the player flips again.

If the probability that game is settled on or after \(8^{\text{th}}\) toss can be written as \[ \left[ a - \frac{b}{c^{d}} \right ]^{e}\] where \(c\) is least positive integer, \(\gcd(b,c)=1\) then find the value of

\(a + b + c + d + e\).

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