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)?
My answer: N+2-1C2-1 = N+1C1. #I am not sure if it is correct
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?
My answer: N+1C1 / NC2. #I am not sure it it is correct
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?
Could you please give some hints?