We flip 100 fair coins 6 times each, how many coins will end up being heads all 6 times or tails all six times? I've solved for a single coin flipped 6 times and the probability that we get 6 flips as all heads or tails as 1/64 --&gt; 1.56%
But I'm stuck on where to go from there.
The binomial probability that a given fair coin will be only heads in 6 trials, or only tails in 6 trials, is $2^{-6}$.  The probability that in 6 trials it is only heads or only talks is thus $2^{-5}$.
If 100 fair coins are flipped 6 times each, the expected number that come up only heads or only tails is therefore $100 \times 2^{-5}=3.125$.
This is equivalent to a non-rounded version of one of the solution proposed by @DontWorry: after 1 roll, you expect $100/2$ coins to be only heads; after 2 rolls, the expectation is $100/2^2$, and after 6 rolls, the expectation is $100/2^6$, so for both only heads or only tails, the expected number is $100/2^5$.
Assuming that
Pr[tails] = 0.5, Pr[heads] = 0.5
Then if we get 100 people flip the coins at the same time and we eliminate the ones with heads we have
100 in the beginning
50 after 1st flip
25 after 2nd flip
12 after 3rd flip (rounding downward/floor)
6 after 4th flip
3 after 5th flip (rounding downward)
1 after 6th flip
So we have in total 1 (or 2 because we rounded downward)  person with tails 6 times in a row
The same for heads
Finally, 1 coin will heads all 6 times and 1 coin will be tails all 6 times.