Question

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

• A. $\frac{1}{6}$
• B. ${}^{12}{C}_2\times \frac{2^6}{{\left(12\right)}^6}$
• C. ${}^{12}{C}_2\times \frac{2^6-1}{{\left(12\right)}^6}$
• D. $\frac{341}{{\left(12\right)}^5}$

Answer 1

Answer: (D)

$\frac{341}{{\left(12\right)}^5}$

Generally any one birthday can fall in one of the $12$ months.
For six different people, the total number of ways is ${\left(12\right)}^6$.
Coming to the favorable ways: first any two months can be chosen in ${}^{12}{C}_2$ ways.
The six birthday can fall in these $2$ months in $2^6$ ways.
Of these the two cases when all the six birthdays fall in one month are to be removed .
$\therefore$ Required probability ${=}^{12}{C}_2\times \frac{(2^6-2)}{{\left(12\right)}^6}=\frac{341}{{\left(12\right)}^5}$