Question 12
For a positive integer n define a sequence of zeros and ones to be balanced if it contains n zeros and n ones. Two balanced sequences a and b are neighbors if you can move one of the 2n symbols of a to another position to form b. For instance, when n = 4, the balanced sequences 01101001 and 00110101 are neighbors because the third (or fourth) zero in the first sequence can be moved to the first or second position to form the second sequence. Prove that there is a set S of at most $ \frac{1}{n+1}\binom{2n}{n} $ balanced sequences such that every balanced sequence is equal to or is a neighbor of at least one sequence in S.