Answer: 495
==============================================================
We want exactly 8 correct answers of false, so we want exactly 4 correct answers of true. In short, we want something like FFFF FFFF TTTT. Because the sequence of T values is shorter than the F values, this means it's a lot easier to count out the number of ways to pick the true responses, rather than the false responses.
Consider we have 4 slots: A, B, C, D
In slot A, we can pick the number 1 through 12 to represent getting a T
For slot B, we have 11 choices because we can't reuse whatever we picked for slot A
Then for slot C we have 10 choices
Finally, slot D has 9 choices
Multiply those values out: 12*11*10*9 = 11,880
If order mattered, then we'd be done with the problem. However, order does not matter. This is because a sequence like {1, 2, 3, 4} is the same as {1,2,4,3}. The only thing that matters is that we have the numbers in the same group. We can rearrange any given group however we want.
Note that there are 4! = 4*3*2*1 = 24 ways to arrange any individual group of four items. So we have to divide the previous result 11,880 by 24 and we get: 11880/24 = 495 which is the answer
If you wanted, you can use the nCr combination formula to get this same answer. You would use n = 12 and r = 4
