The probability that the birth days of six different persons will fall in exactly two calendar months is

$\frac{341}{12^5}$

Since anyone birthday can fall in one of 12 months. So, total number of elementary events = 12⁶.

Now, any two months can be chosen in ${^{12}C}_2$ ways. The six birth days can fall in these two months in 2⁶ ways. Out of these 2⁶ ways there are two ways when all the six birth days fall in one month. So,

Favourable number of elementary events = ${^{12}C}_2 × (2^6 - 2)$

Hence, required probability = $\frac{^{12}C_2 × (2^6 - 2)}{12^6}$

$=\frac{12×11×(2^5-1)}{12^6}=\frac{341}{12^5}$