'possible outcomes' definition and interpretation I am baffled on a practice problem I am doing:
"A fair coin is flipped 25 times, what are the total possible outcomes?"
My question is how do we define and interpret 'possible outcomes'? 
Interpretation 1 : 25 + 1 = 26 possible outcomes, as in counts of heads and tails. So 1 heads and 24 tails is an outcome, and then 2 heads and 23 tails is another outcome. 
Interpretation 2 : As a permutation with repetitions allowed = (26 ^ 2); So in this case, even if the count of heads and tails is the same, if they have a different order, it will count as a different 'possible outcome'.
How is it generally accepted to interpret this problem?
Thank you. 
 A: The correct interpretation is probably "possible sequences of heads and tails." (I believe this is what you meant by your second interpretation, but you've incorrectly calculated it.) Each flip has $2$ possible results, and there are $25$ flips, so there are $2^{25}$ possible outcomes.
To see where that comes from, let's consider some cases with fewer flips. If I flip the coin only once, then there are only $2=2^1$ possible outcomes. Flipping it twice doubles the possible outcomes, since I can still flip heads or tails, regardless of the first flip, so there are $4=2^2$ possible outcomes with two flips. More generally, if there are $2^{n-1}$ possible outcomes for $n-1$ flips, then flipping $n$ times again doubles the possible outcomes by the same reasoning, so there are $2\cdot 2^{n-1}=2^n$ possible outcomes with $n$ flips. By induction this holds for all $n$.
As for why I say that the second interpretation is probably what is intended, note that if our outcomes are the possible sequences, then each count (such as "$4$ heads are flipped") is an event. However, if our outcomes are counts, then we cannot (from those outcomes) say anything about particular types of sequences (such as "the first $3$ flips are heads"). We would like for our outcomes to be, in a sense, fundamental--to be able to tell us basically anything about the experiment that we might want to know. The second interpretation allows for that. The first does not.
