which I wasted a bunch of time searching trying to figure out why that was wrong). And you were right that calculating the CDF is impractical.

I think, based on the AP formula sheet (I'm rusty, this might not be a full credit answer...)
https://apcentral.collegeboard.org/pdf/statistics-formula-sheet-and-tables-2020.pdf
they want her to know that for a large enough sample size a normal distribution approximates a binomial distribution. So she would calculate the estimated mean and standard deviation:
u = n*p = 500 * .47 = 235
o = sqrt(n*p(1-p) = 11.16

So, in this context we want to know the p(x > 251), so first, calculate a z-score.
z = (x - u) / o = 251-235 / 11.16 = 1.43, looking at the table it's .9236 for P(x < 251), so
P(x > 251) = 1 - .9263= .0764

Plugging it into https://www.gigacalculator.com/calculators/binomial-probability-calculator.php shows P(x > 250) = .0825

So, uh, hopefully this jogs your daughter's memory...