Hmmm, this is a tough one. However, I do not believe the statement in (a), "there are n! linear orderings of {1, 2,...,n}", is true. Why would you try to prove it so?
Is that the case, that you cannot prove this? The questions were given as a homework assignment, so I'm assuming that there is a way to prove this. It works when you do the arithmetic. For example, 2! = 2 and there are two ways to order 1 and 2: {1,2} and {2,1}. This is as far as I've gotten.