Clarifying an old question on Combinations. So there is already a question here and I just want to clarify something in this.
Link:Meaning of the question
Now the accepted answer says that the answer is a power of 2.And there is an explanation of that in another answer but I could not understand that.Could someone please explain the 2 power to me.I would be really grateful for that as this question is stuck in my mind.
The question for those in a hurry:"The supreme court has given a 6 to 3 decisions upholding a lower court; the number of ways it can give a majority decision reversing the lower court is "
 A: Each judge can vote for or against. Thus each judge has two choices, independent of all other judges. Thus, by the multiplication principle, the $9$ judges can vote in $2^9$ different ways.
Since there is an odd number of them, there can't be a tie. However they vote, there's a majority either for or against. By flipping all $9$ votes, you get a one-to-one correspondence between the votes with majority for and the votes with majority against. Since the two are in one-to-one correspondence, there must be the same number of each, and thus the number of each must be half the total. Thus the number of ways the judges can vote to overturn the decision is $2^9\div2=2^8$.
Edit in response to the comment: Imagine some voting pattern of the judges, say, YYNYNNYY (where the judges are arranged in a row in some arbitrary but fixed order). Now flip all the votes, yielding NNYNYYNN. This flipped pattern has a majority for if and only if the original pattern had a majority against, and vice versa. If you flip the flipped pattern, you recover the original pattern. Thus, the set of all voting patterns decomposes into pairs of patterns that are transformed into each other by the flipping operation. Each pair contains exactly one pattern with majority for and one pattern with majority against. Thus half the patterns must have majority for and half must have majority against.
