The N students need to be divided into two groups: X , and Y . However, they have several issues with how they want to be arranged.

The N students need to be divided into two groups: X, and Y. However, they have several issues with how they want to be arranged. In particular, two students, student A and student B, really enjoy working together, and would like to be in the same group. This is not always possible, as the teaching team wants to fairly arrange the students, and not favor any particular pair or group. Please help them divide up the class. 

First, arrange the N students in a line. What is the probability that students A and B are next to each other?

Now, arrange the N students in a circle. What is the probability that students A and B are next to each other?

For Part C and D, assume that it is equally likely for a given student to be born in any of the 12 months.

What is the probability that students A and B do not have birthdays in the same month?

What is the probability that none of the students share the same birthday month?

How many different sizes of group X are possible (including 0)?

If each student is equally likely to be put into either group X or group Y, what is the probability that group X has exactly 2 people?

If after dividing all of the students into groups X and Y, group X has exactly 2 people, what is the probability that those two people are students A and B?

If after dividing all of the students into groups X and Y, group X has exactly R students, what is the probability that students A and B are both in group X?