Question

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

• A. $\frac{134}{{12}^5}$
• B. $\frac{143}{{12}^5}$
• C. $\frac{341}{{12}^5}$
• D. $\frac{331}{{12}^5}$

Answer 1

The correct option is C $\frac{341}{{12}^5}$

Since anyone's birthday can fall in one of $12$ months, so total number of ways is ${12}^6$.

`Now, any two months can be chosen in ${}^{12}{C}_2$ ways. The six birthdays can fall in these two months in $2^6$ ways. Out of these $2^6$ way there are two ways when all the six birthdays fall in one month.
So, favourable number of ways is ${}^{12}{C}_2\times (2^6-2).$
Hence, required probability is

$\frac{{}^{12}{C}_2\times (2^6-2)}{{12}^6}=\frac{12\times 11\times (2^5-1)}{{12}^6}=\frac{341}{{12}^5}$