Given: The Supreme Court has given a 6 to 3 decision upholding a lower court.

To find: The number of ways it can give a majority decision reversing the lower court.

Solution:

When the Supreme Court upholds a lower court's decision, it means that they agree with the lower court. However, if the Supreme Court wants to reverse the lower court's decision, they need to have a majority decision in favor of the reversal.

Let's assume that the 6 judges in the Supreme Court who voted to uphold the lower court's decision are A1, A2, A3, A4, A5, and A6. Similarly, the 3 judges who voted against the lower court's decision are B1, B2, and B3.

To reverse the lower court's decision, the Supreme Court needs at least 5 judges to vote in favor of the reversal. Let's consider the following cases:

Case 1: 5 judges vote in favor of reversal

In this case, we need to select 5 judges out of the 9 judges available. The remaining 4 judges will automatically be in favor of upholding the lower court's decision. Therefore, the number of ways to select 5 judges out of 9 is:

9C5 = 126

Case 2: 6 judges vote in favor of reversal

In this case, we need to select 6 judges out of the 9 judges available. The remaining 3 judges will automatically be in favor of upholding the lower court's decision. Therefore, the number of ways to select 6 judges out of 9 is:

9C6 = 84

Case 3: 7 judges vote in favor of reversal

In this case, we need to select 7 judges out of the 9 judges available. The remaining 2 judges will automatically be in favor of upholding the lower court's decision. Therefore, the number of ways to select 7 judges out of 9 is:

9C7 = 36

Case 4: 8 judges vote in favor of reversal

In this case, we need to select 8 judges out of the 9 judges available. The remaining 1 judge will automatically be in favor of upholding the lower court's decision. Therefore, the number of ways to select 8 judges out of 9 is:

9C8 = 9

Case 5: 9 judges vote in favor of reversal

In this case, all 9 judges are in favor of reversing the lower court's decision. Therefore, there is only one way to select all 9 judges.

Total number of ways to give a majority decision reversing the lower court = 126 + 84 + 36 + 9 + 1 = 256

Therefore, the correct answer is option A) 256.

⁹C5 + ⁹C6 + ⁹C7 + ⁹C8 + ⁹C9
after solving this we get
126+84+36+9+1=256