Problem Statement  
John is playing with balls. All of the balls are identical in weight and considered to have a zero radius. All balls are located on the same straight line and can move only along this line. If a ball rolling to the right and a ball rolling to the left at the same speed collide, they do not change speed, but they change direction.
You are given int[] x. x[i] is the initial position of the ith ball. John decides the direction for each ball (right or left) with equal probability. At time 0, he rolls the balls in the chosen directions simultaneously at a speed of one unit per second. Return the expected number of bounces between all balls during T seconds (including those collisions that happen exactly at T seconds).  
Definition  
 
Notes  
  There is no friction. Each ball continues rolling at the same speed forever.  
  Your return value must have an absolute or relative error less than 1e9.  
Constraints  
  x will contain between 1 and 12 elements, inclusive.  
  Each element of x will be between 0 and 100,000,000, inclusive.  
  All elements of x will be distinct.  
  T will be between 1 and 100,000,000, inclusive.  
Examples  
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