Question

The probability that out of 10 persons, all born in April, at least two have the same birthday is

• A. $\frac{{}^{30}{P}_{10}}{(30{)}^{10}}$
• B. $1-\frac{{}^{30}{P}_{10}}{30!}$
• C. $\frac{{30}^{10}{-}^{30}{P}_{10}}{{30}^{10}}$
• D. $noneofthese$

Answer 1

The correct option is C $\frac{{30}^{10}{-}^{30}{P}_{10}}{{30}^{10}}$

There are 30 days in April but each person can have birthday in 30 days.
$\therefore n\left(S\right)=30\times 30\times ...\times 30\left(10times\right)={30}^{10}$
n(E) = n(S) - The number of ways in which 10 persons can have different birthdays.
$\therefore n\left(E\right)={30}^{10}{-}^{30}{P}_{10}$
$\therefore P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}=\frac{{30}^{10}{-}^{30}{P}_{10}}{{30}^{10}}$