Probability of being picked more than once in a simple random sample. A school has $500$ girls and $500$ boys. A simple random sample is obtained by selecting names from a box (with replacement) to a get a sample of $10$. 
Find the probability of someone being picked more than once.
My working is: 
\begin{align}
P(\text{being picked more than once}) &= 1 - P(\text{not being picked}) - P(\text{being picked exactly once})\\
&=1 - \left(\frac{999}{1000}\right)^{10} - 10\left(\frac{1}{1000}\right)\left(\frac{999}{1000}\right)^9 = 0.00004476
\end{align}
However, this is not correct. Any ideas?

Probability of anybody being picked more than once:
$$
\frac{\binom{10^{3}}{10}\times 10!}{\left(10^{3}\right)^{10}}
$$
Probability of a particular person being picked more than once:
$$
1-\frac{\left(10^{3}-1\right)^{10}+10\times\left(10^{3}-1\right)^{9}}{\left(10^{3}-1\right)^{10}}
$$
