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There is a question that goes like this :
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 :

Actually in this question, I can't get the statement what does the statement "6 to 3 decisions upholding a lower court" mean and then what we have to find ?

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    $\begingroup$ There are 9 justices on the U.S Supreme Court. 6 voted in favor of the decision and 3 voted against. $\endgroup$ Jun 8, 2011 at 6:42
  • $\begingroup$ I say the "6 to 3" clause is irrelevant; all you're being asked for is how many ways the court can give a majority vote in favor of a particular option. $\endgroup$ Jun 8, 2011 at 6:51

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The supreme court has 9 members. A "6 to 3" decision means that 6 judges voted with the lower court, and 3 voted against.

To reverse the order, a majority of judges must vote to reverse it. So either 5, 6, 7, 8, or 9 judges must vote to reverse the decision. The question is, how many ways can this happen?

Does that make sense?

HINT: The answer is a 2 power (but that's deceptively non-obvious from the sum).

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    $\begingroup$ on the other hand, it's blindingly obvious that it's a power of $2$ when you consider the total number of ways to vote and the one-one correspondence between votes that uphold and votes that reverse. $\endgroup$ Jun 8, 2011 at 6:50
  • $\begingroup$ @Gerry - oh, that's true. ;p $\endgroup$
    – davidlowryduda
    Jun 8, 2011 at 6:51
  • $\begingroup$ ok..I got it. So it must be 9C5 + 9C6 + 9C7 + 9C8 + 9C9. So is the statement "The supreme court has given a 6 to 3 decisions upholding a lower court" is used only to determine total number of judges ? It has nothing to do with rest of the question...right ? $\endgroup$ Jun 8, 2011 at 6:52
  • $\begingroup$ @Happy: yeah... not every question is a good one... $\endgroup$
    – davidlowryduda
    Jun 8, 2011 at 6:56
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Got it . So There are 9 judges in total and 6 are voting for the decision of lower court and 3 are voting against . So in how many ways the decision can be reversed .

It can by 5 votes for or 6,7,8 and the whole 9 to get majority so ,

9C9+9C8+9C7+9C6+9C5

=1+9+36+84+126 =256 is the answer

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Since there are $9$ judges, there is no case of a tie (equal number of votes for and against). Thus, the number of ways that "for" gets a majority is equal to the number of ways that "against" gets a majority, and therefore, each must be half of the total number of ways to vote.

The total number of ways to vote is, of course, $2^9 = 512$ (as there are nine judges and each independently votes for or against).

So, the answer is ${256}$.

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