If language L₁ is regular and language L₂ is regular, which of the following is/are also regular?

a. L₁ ? L₂

b. L₁ ? L₂

c. L₁L₂

d. L₁*

e. all of the above

The language of strings that start with a and have 4 b's and have length < 8 is regular.

Select one:

a. true

b. false

I know this one is true since I already got it wrong, but why is it true? I thought Finite State Machines cannot store or count strings and any problem that requires it is not a regular language. How would you draw a DFA that gives the choice between a or b after the first a and still verify there's 4 b's somewhere mixed in there without enforcing them at specific locations?