Combinatorics Fever

In each game there is a $\dfrac{20}{2^{10}}$ chance that exactly one "odd" flip will come up. Why? The denominator is $2^{10}$ because there are $2$ choices for each coin (heads or tails) and $10$ independent flips. The numerator is $20$ because there are $10$ different people who could win, and 2 choices for what the odd flip is (heads or tails).

Now in order for the game to be settled on the $8^{th}$ flip or higher, that means the first 7 games do not have exactly one odd flip. The probability of this can be written as: $\left(1 - \dfrac{20}{2^{10}}\right)^7 = \left(1 - \dfrac{5}{2^{8}}\right)^7$ Therefore, we have $a+b+c+d+e = \boxed{23}$ .