Assign n professors to 2n courses in two semesters and in spring no professor teaches the same pair of course So the problem is


  A math department has n professors and 2n courses, two assigned to each professor each
  semester. How many ways are there to assign the courses in the fall semester? How many
  ways are there to assign them in the spring so that no professor teaches the same two
  courses in the spring as in the fall? If all the assignments are equally likely, what is the probablity of this event. 


For the first part, I believe it is 2n!/(2)**n. But I keep having trouble wrapping my head around the second and third one. I was thinking if there are no professor teaches the same course, then it is a lot like getting them assigned first, and ask each professor to switch course with another professor. So the first one has (n-1) choice, second one has (n-2) choice, and it will end up being (n-1)!, and by rule of product, there are [2n!/(2)**n]*(n-1!). But I feel that this number might be too big, and I might be double counting something. And for the third question, I think the denominator would be [2n!/(2)**n]**2 by rule of product. But I don't know what to do with nominator. 

Thanks in advance!
Part II:

Using the principle of Inclusion and Exclusion,

you get then number of ways to be $ = \sum_{0}^{n} (-1)^n {n\choose k} {2n-2k\choose 2,2,2..}$ k number of 2's.

ofcourse for n= 1, the number of ways$=0$ as @Bram28 has asked you to confirm.

The answer for part III is $\frac{Part II Answer}{Part I answer}$ that is numerator is the number of ways the same courses are not taught by the professor in Spring and the denominator is the total number of ways courses could be assigned in a semester.
Part II:

Using the principle of Inclusion and Exclusion,

you get then number of ways to be $ = \sum_{0}^{n} (-1)^n {n\choose k} {2n-2k\choose 2,2,2..}$ k number of 2's.

ofcourse for n= 1, the number of ways$=0$ as @Bram28 has asked you to confirm.

The answer for part III is $\frac{Part II Answer}{Part I answer}$ that is numerator is the number of ways the same courses are not taught by the professor in Spring and the denominator is the total number of ways courses could be assigned in a semester.