a. Doesn't matter
Almost everyone, including top mathematicians, will say it doesn't matter; flipping coins is 50/50, either pattern can hit first.
The answer is C
Heads-tails-heads will average 10 flips to hit, and heads-tails-tails will average 8 flips to hit.
This is important. It shows us how incompetent we are when it comes to logically thinking about probability.
Unfortunately, probability assessment is important in modern life, where we live among billions and have instant communication globally. Rare things happen, and we all hear about them, but don't have any idea why it seems like planes are falling out of the sky every day but it's still safe to fly.
It becomes even more important in courtrooms.
In a famous case in the UK, a well regarded pediatrician told a jury that the chances of having SIDS twice in one family was 1 in 73 million. He arrived at this figure by finding that the chance of SIDS occurring was 1 in 8,500 -- so he simply multiplied 8,500 by 8,500 and arrived at 1 in 73 million.
His calculations are completely wrong, yet went unchallenged by the defense. His calculation is only part of the story. We have no idea was the real probability is, because we don't know that SIDS is randomly distributed, how reliable the diagnosis is, or if there genetic factors, etc.
Thinking that the chance is 1 in 73 million is exactly the same as believing that a 99% accurate cancer test coming up positive means the patient has a 1 in 100 chance to be cancer free. As per my last two notes, we know this is not the case. We need to know how many people actually have cancer, and compare it to our personal test results.
Similar misunderstandings lead the way for the popularity of lotteries and casino gambling. We're all just idiots when it comes to big numbers and happenstance.