Depends how you define advanced.

I teach the binomial distribution to final year high school students here in Oz. They seem to understand it.

Your problem is binomial with p=0.15 and n=25. Let X be the occurrance of a serious injury.

Note

For the first question find P(X=5)

For the second find P(X=2)+P(X=3)+...+P(X=8)

For the third find P(X=0)

For the forth P(X>0) = P(X=1)+P(X=2)+...+P(X=25) or even easier 1-P(X=0)

Does this help?