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Given: Nine persons A, B, C, D, E, F, G, H, and I
Celebrated their birthdays in different months of the same year which are January, February, March, April, May, June, July, August, September.
January, March, May, July and August being the months having 31 days.
1) F celebrates his birthday in the month which has 30 days but before June.
2) Three persons celebrate birthday between F and E. E is exactly between G and C, and G is not the last person to celebrate birthday.
3) B is the first person to celebrate birthday.

4) Two persons celebrate birthday between A and H, and A is not an immediate neighbour of B. But he celebrates his birthday in the month which has 31 days.

5) Two persons are there between D and I and D celebrates birthday before I. I is not celebrating after H.

Thus, all the four option pairs are the month of 31 days except April - September.
Hence, the correct answer is "April-September".