Question

The probability that the birthdays of 6 boys will fall exactly in two calender months is

• A. $\frac{{}^{12}{C}_2}{{12}^6}$
• B. $\frac{{}^{12}{C}_2\times 2^6}{{12}^6}$
• C. $\frac{{}^{12}{C}_2\times (2^6-2)}{{12}^6}$
• D. $\frac{{}^{12}{C}_2\times (2^6-3)}{{12}^6}$

Answer 1

Answer: (C)

$\frac{{}^{12}{C}_2\times (2^6-2)}{{12}^6}$

Total cases here is : ${12}^6$ as each boy has 12 choices and 6 boys.
Now if we want them to have their birthdays in exactly 2 months, we need to choose the 2 months first. this can be done in ${}^{12}{C}_2$ ways.
Do this and then each boy has 2 choices, month A or month B. $⟹2^6$
but we also need to subtract the cases in which all of them have their birthdays in one month exactly. So we subtract $2$ cases.
Hence probability$=\frac{{}^{12}{C}_2\times (2^6-2)}{{12}^6}$