Question

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

• A. $\frac{{{}^{3}C}_{10}}{{\left(30\right)}^{10}}$
• B. $1-\frac{{{}^{3}C}_{10}}{30!}$
• C. $\frac{{\left(30\right)}^{10}-{{}^{30}C}_{10}}{{\left(30\right)}^{10}}$
• D. None of these

Answer 1

The correct option is C $\frac{{\left(30\right)}^{10}-{{}^{30}C}_{10}}{{\left(30\right)}^{10}}$

Total no. of ways of 10 persons birthday in April(30days)= ${\left(30\right)}^{10}$

Total ways of single birthday, single day= ${}^{30}{C}_{10}$

Total ways of atleast two have same birthday = Total ways - single days

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

Probability of atleast two have the sme birthday is

=$\frac{Atleasttwobirthday}{Totalways}$

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