First, find the probability of scoring higher than 78. Scores are normally distributed, so you have
\(\mathbb P(X>78)=\mathbb P\left(\dfrac{X-68.2}{10.4}>\dfrac{78-68.2}{10.4}\right)\approx\mathbb P(Z>0.9423)\approx0.173\)
Now, the event that any given student scores higher than 78 follows a binomial distribution. Here you have 75 total students (so \(n=75\)) with success probability \(p=0.173\).
So the probability of getting 20 students that fit the criterion is
\(\mathbb P(Y=20)=\dbinom{75}{20}p^{20}(1-p)^{75-20}\approx0.0134\)
