case 1: What is the number of bit strings of length 4, with 1 zero and 3 ones, zero must be followed by one

Answer: 3

case 2: What is the number of bit strings of length 6, with 2 zeros and 4 ones, zero must be followed by one

Answer: 6

Question: What is the number of bit strings of length n, with n/2-1 zeros and n/2+1 ones, zero must be followed by one (where n is even and n>2)

Answer: ???

Observation from the problem are: they all end with one, and no 2 consecutive zeros can occur

W/e you have in mind either recursion, closed form, permutation/combination or else anything helps.