$\mathbf{Question:}$ What is the probability of $1$st January being the birthday of $two$ people when chosen (two persons are picked randomly) among $500$ people? ($365$ days in a Year).
$\mathbf{Attempt:}$ Let us denote the persons by $p_i$, $1\leq i \leq 500$. We write the birthdays as the "$n$ th" day of the year. The set of all possibilities:
$S=\{\{p_1,1\},\{p_1,2\},...,\{p_1,365\},\{p_2,1\},\{p_2,2\},...,\{p_2,365\},...,\{p_{500},1\},\{p_{500},2\},...,\{p_{500},365\} \}$.
$|S|=500 \times 365=182500$. Denote $\{p_i,j\}\equiv x_{i,j}$, $1\leq j\leq 365$.
Now, when we pick two persons at random, the total number of possibilites become : $\displaystyle{{182500}\choose{2}} -500 \times {365 \choose 2}=N$ (say). [Explanation: When we pick $u=\{p_1,1\}$, we cannot choose another possible value of $p_1$. So the other choice is from the rest of the elements of $S$, which amounts to $182500-365=182135$ possible pairs involving $u$. Continuing, we get $182500*182135/2=N$].
Now, the number of possible outcomes when picked two at a time from the set $\{\{p_1,1\},\{p_2,1\},...,\{p_{500},1\}\}$ is ${500 \choose 2}=P$ (say).
Therefore, the answer is $P/N=\displaystyle \frac{1}{133225}$.
Is this okay? Kindly verify.