# Clarifying an old question on Combinations.

So there is already a question here and I just want to clarify something in this.

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 "

• I want to know how to solve it by understanding the powers and not as $^9C_5+^9C_6+^9C_7+^9C_8+^9C_9$ May 31, 2016 at 7:53

## 1 Answer

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.

• Sorry to be rude but this answer is exactly like the one given in the original question(but was not accepted as it is answered later).I did not understand the one to one correspondence. (I have studied only till High School yet and preparing for examinations) May 31, 2016 at 8:01
• @IshanTaneja: You made no attempt whatsoever in the question to explain what part you didn't understand. I tried as best I could to explain all parts in a bit more detail. If you want yet more detail, you'll have to invest some effort into pointing out what parts you don't understand. I'll explain the one-to-one correspondence in more detail. May 31, 2016 at 8:02
• Ah now I get it.Really helpful.Also what would be the case if there are 10 judges,any hint how the solution would go. (If it is too long you can ignore this, it is not relevant to this question ,just a thought) May 31, 2016 at 8:12
• @IshanTaneja: In that case, you'd have to take the possibility of a tie into account. There are $\binom{10}5$ ways to choose $5$ of the $10$ judges who vote against, so this is the number of ways to have a tie. All remaining voting patterns again fall into pairs, so in this case the number of ways to vote to overturn would be $\frac{2^{10}-\binom{10}5}2=2^9-\frac12\binom{10}5=512-\frac12\cdot252=386$. May 31, 2016 at 8:15