Let x_{1}, x_{2}, \ldots, x_{k} be a sequence of integers. A rearrangement of this sequence (the numbers in the sequence listed in some other order) is called a scramble if no number in the new sequence is equal to the number originally in its location. For example, if the original sequence is 1,3,3,5 then 3,5,1,3 is a scramble, but 3,3,1,5 is not.
A rearrangement is called a two-two if exactly two of the numbers in the new sequence are each exactly two more than the numbers that originally occupied those locations. For example, 3,5,1,3 is a two-two of the sequence 1,3,3,5 (the first two values 3 and 5 of the new sequence are exactly two more than their original values 1 and 3 ).
Let n \geq 2. Prove that the number of scrambles of
is equal to the number of two-twos of
(Notice that both sequences have n+1 numbers, but the first one contains two 1 \mathrm{~s}.)