
    Design a function for an automated grading system that calculates how many possible true/false answer combinations exist for a test with 2n questions, where students must have the same number of "true" responses in the first n questions as in the last n questions. The system uses this to detect balanced response patterns and potential knowledge gaps across test sections.
    

    
def count_binary_seq(n):
    """
    Design a function for an automated grading system that calculates how many possible true/false answer combinations exist for a test with 2n questions, where students must have the same number of "true" responses in the first n questions as in the last n questions. The system uses this to detect balanced response patterns and potential knowledge gaps across test sections.
    

    """
def count_binary_seq(n):
    """
    Design a function for an automated grading system that calculates how many possible true/false answer combinations exist for a test with 2n questions, where students must have the same number of "true" responses in the first n questions as in the last n questions. The system uses this to detect balanced response patterns and potential knowledge gaps across test sections.
    

    """
    nCr = 1
    res = 1
    for r in range(1, n + 1):
        nCr = nCr * (n + 1 - r) / r
        res += nCr * nCr
    return res