Detailed explanations please. Thank you.

A)We roll a fair coin 43 times. Let X be the total number of trials when two consecutive outcomes coincide. a) Find P(X = 12). b) Find E(X + 1)².

B) Suppose that n+2 identical balls are randomly distributed into n distinct boxes. Given that no box is empty, find the probability that box 1 contains 3 balls.

C) Suppose that 105 identical balls are randomly distributed into 100 distinct boxes. Given that the total number of balls in the first three boxes is 6, find the probability that no box is empty.

D) Suppose that 2012 identical balls are randomly distributed into 2010 distinct boxes. Given that no box is empty, find the probability that box 1 contains more balls than box 2.