The probability that the birthdays of 4 different persons will fall in exactly two calendar months is :

Each of the 4 people can have a birthday in any of the 12 calendar months. The total number of ways the birthdays can be distributed is = $$12\times\ 12\times\ 12\times\ 12=12^4$$

Now, for the birthdays to fall in exactly two calendar months, we must first choose which two months these will be. The number of ways of choosing = $$^{12}C_2=\dfrac{12\times\ 11}{2}=66$$

For each person, their birthday can be in one of the two chosen months. This gives $$2^4=16$$ total possibilities.

However, this includes two cases where all four birthdays fall in just one of the two months (either all in the first chosen month or all in the second). We must subtract these two cases to ensure the birthdays fall in exactly two months.

So, number of ways = $$2^4-2=16-2=14$$ possibilities.

Total number of favourable cases = $$66\times\ 14$$

So, required probability = $$\dfrac{66\times\ 14}{12\times\ 12\times\ 12\times\ 12}=\dfrac{77}{1728}$$

So, option A is the correct answer.