Question

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

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

Answer 1

Answer: (D)

$\frac{{30}^{10}{-}^{30}{C}_{10}}{{\left(30\right)}^{10}}$

The problem states that at least two people may have the same birthday which we simply exclude all cases where everyone has a different birthday

Total cases ${30}^{10}$ as everyone can be form on any of the days

Unfacourable cases ${}^{30}{C}_{10}$ as in the cases each person is born on different days of the month

Favourable cases=TOtal-Unfacourable cases $={30}^{10}-{}^{30}{C}_{10}$

Probability $\frac{Favourablecases}{Totalcases}=\frac{{30}^{10}-{}^{30}{C}_{10}}{{30}^{10}}=\left(D\right)$