Step 1: Recall the discrete versus continuous divide.
A distribution is continuous when its random variable can take any value inside an interval, and discrete when the variable can only take separated, countable values such as whole numbers. This is the only property that needs to be checked here.
Step 2: Sort each named distribution by this rule.
Binomial distribution counts the number of successes out of a fixed number of trials, so its outcomes are whole numbers only, making it discrete. Poisson distribution counts the number of events occurring in a fixed interval, which is also always a whole number, so it is discrete as well.
Step 3: Confirm the continuous group.
Normal distribution models a variable that can take any real value on the number line. Chi-square distribution is built from sums of squares of continuous normal variables, and F-distribution is a ratio of two such Chi-square variables, so all three, A, D and E, are continuous, leaving B and C as the discrete pair.
\[ \boxed{\text{(A), (D) and (E) only.}} \]