Question 1205509
answer is selection D.
i did it two ways.
first way was direct, using the binomial distribution formula, in excel.
that formula is p(x) = p^x * q^(n-x) * c(n,x)
all values of p(10) to p(42) are summed up to get the probability of x being greater than or equal to 10.
the probability of x being greater than or equal to 10 was equal to  0.324376129
i then used normal approximation to the binomial.
p(x >= 10) = selection D =  0.3357
the normal approximation won't be exact, but it'll be close.


here are the results using the normal distribution calculator at <a href = "https://davidmlane.com/hyperstat/z_table.html" target = "_blank">https://davidmlane.com/hyperstat/z_table.html</a>
here are the results.


<img src = "http://theo.x10hosting.com/2024/010201.jpg">


the mean was set at 10
the standard deviation was set at sqrt(42 * .2 * .8) = 2.592296


here is a reference on normal approximation of the binomial.
<a href = "https://stats.libretexts.org/Courses/Las_Positas_College/Math_40%3A_Statistics_and_Probability/06%3A_Continuous_Random_Variables_and_the_Normal_Distribution/6.04%3A_Normal_Approximation_to_the_Binomial_Distribution" target = "_blank">https://stats.libretexts.org/Courses/Las_Positas_College/Math_40%3A_Statistics_and_Probability/06%3A_Continuous_Random_Variables_and_the_Normal_Distribution/6.04%3A_Normal_Approximation_to_the_Binomial_Distribution</a>


with proportions, the mean is equal to n * p.
in this case it was 42 * .2 = 8.4
the standard error is equal to sqrt(n * p * q) = sqrt(42 * .2 * .8).


let me know if you have any questions.
theo