I got a different answer, Melody.
If a number is both a square and a 5th power, it must also be a perfect 10th power.
Note that \(1024^2 = 4^{10}\), so there are 4 numbers that work: \(1^{10}\), \(2^{10}\), \(3^{10}\), and \(4^{10}\)
There are \({4 \choose 2} = 6\) successes and \({1024 \choose 2} = 523776\) total outcomes.
So, the probability is \({6 \over 523776} = \color{brown}\boxed{1 \over 87296}\)
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