The numbers can be formed using the digits $$1,2,3,4,3,2,1$$ so that odd digits always occupy odd places in __________ ways.

For a set of five true or false questions, no student has written the all correct answers and no two students have given the same sequence of answers. The maximum number of students in the class for this to be possible is

The number of ways in which 8 different pearls can be arranged to form a necklace is

If $$\frac{n !}{2 !(n-2) !}$$ and $$\frac{n !}{4 !(n-4) !}$$ are in the ratio $$2: 1$$, then $$n=$$