Question

The probability that the birthday of six different people will fall in exactly two calendar months which are not consecutive is

• A. $\frac{1}{6}$
• B. ${}^{12}{C}_2.\frac{2^6}{{12}^6}$
• C. $\frac{55\ast 31}{{12}^5\ast 6}$
• D. $\frac{55}{6^6}$

Answer 1

Answer: (C)

$\frac{55\ast 31}{{12}^5\ast 6}$

Generally any one birthday can fall in anyone of the 12 months. Therefore the total number of ways is ${12}^6$ .\
For the favorable event any two months can be chosen in ${}^{12}{C}_2$ ways.
Since the month are not be consecutive , the two months can be chose in $({}^{12}{C}_2-11)$ ways.
The six birthday can fall in these two months in $2^6$ . of these two cases (when all the six birthday fall in one month) are to be removed .
Probability $\frac{{(}^{12}{C}_2-11).(2^6-2)}{{12}^6}=\frac{55\ast 31}{{12}^5\ast 6}$