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.
Q1. How many different sizes of group X are possible (including 0)?
Q1.1 If each student is equally likely to be put into either group X or group Y, what is the probability that group Xhas exactly 2 people?
Q1.2 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?