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$\begin{align*} &X = \text{No of people having same month birth day} \\ &\text{ We need }, P(X \geq 2) \geq \frac{1}{2} \\ \\ \hline \\ \\ &\Rightarrow 1 - P(X \leq 1) \geq \frac{1}{2} \\ &\Rightarrow 1 - P(\text{No one having same month birth day}) \geq \frac{1}{2} \\ &\Rightarrow P(\text{No one having same month birth day}) \leq \frac{1}{2} \;\;\;\; \dots (1)\\ \\ \hline \\ \\ &\text{Now, } k = \text{Minimum no of people required to satisfy } (1) \\ &\Rightarrow P(\text{No one having same month birth day}) = \frac{11*10*9 \dots *(12-k+1)}{12*12*12 \dots *12} \\ &\Rightarrow \text{for k = 4 } \;\; P(\text{No one having same month birth day}) = \frac{11.10.9}{12.12.12} = 0.572 \geq 0.5\\ &\Rightarrow \text{for k = 5 } \;\; P(\text{No one having same month birth day}) = \frac{11.10.9.8}{12.12.12.12} = 0.38 \leq 0.5\\ \end{align*}$

 

Answer $5$.
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