Question

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

• A. $\frac{{}^{30}{C}_{10}}{(30{)}^{10}}$
• B. $1-\frac{{}^{30}{C}_{10}}{30!}$
• C. $\frac{(30{)}^{10}{-}^{30}{P}_{10}}{(30{)}^{10}}$
• D. none of these

Answer 1

Answer: (C)

$\frac{(30{)}^{10}{-}^{30}{P}_{10}}{(30{)}^{10}}$

Probability that all are born on different days = $\frac{{}^{30}{P}_{10}}{{30}^{10}}$
Thus, probability that at least $2$ people are born on the same day = $1-\frac{{}^{30}{P}_{10}}{{30}^{10}}=\frac{{30}^{10}{-}^{30}{P}_{10}}{{30}^{10}}$