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

Since anyone birthday can fall in one of 12 months. So, total number of elementary events = ${12}^6$
Now any two months can be chosen ${ }^{12}{C}_2$  ways. The six birth days can fall in these two months in $2^6$  ways. Out of these $2^6$ ways there are two ways when all the six birthdays fall in one month. So, Favorabale number of elementary events =  ${ }^{12}{C}_2\times (2^6-2)$
Hence, required probability = $\frac{{ }^{12}{C}_2\times (2^6-2)}{{12}^6}=\frac{12\times 11\times (2^5-1)}{{12}^6}=\frac{341}{{12}^5}$