Why does Solve work on a sum with 34 terms but not 35 terms?

Questions
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1

Why does this work when M=34 but not when M=35?:

Clear [n, M, confidence, k, p];
n = 8;
M = 34;
confidence = 0.5;

$ \text{Solve}\left[\sum
_{k=n+1}^M \frac{M! p^k
(1-p)^{M-k}}{(M-k)!
k!}=\text{confidence},p,\mathbb{R}\right] $

2

This code works on my system:

With[{M = 35, n = 8, c = 1/2}, 
 Solve[Sum[(M! p^k (1 - p^(M - k)))/((M - k)! k!), {k, n + 1, M}] == 
   c, p, Reals]]
3

Thanks!!