Question

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

• A. $\frac{77}{1728}$
• B. $\frac{17}{121}$
• C. $\frac{11}{144}$
• D. none of these

Answer 1

The correct option is A $\frac{77}{1728}$

Since a person's birthday can fall in any of the 12 months, So, total number of ways $={12}^4.$
Now, any two months can be chosen in ${}^{12}{C}_2$ ways. The 4 person birthday can fall in these two months in $2^4$ ways. Out of these $2^4$ ways, there are two ways when all of the four birthdays fall in one month.
So, favourable number of ways $=\frac{{}^{12}{C}_2\times (2^4-2)}{{12}^4}=\frac{77}{1728}$