Step 1

Each of three successive vehicles can go in three directions, denote

R - vehicle turns right on a particular freeway exit;

L - vehicle turns left on a particular freeway exit;

S - vehicle goes straight on a particular freeway exit.

For example, event [RLS] would mean that the first vehicle turned right, second vehicle turned left and third vehicle went straight.

a): All possible outcomes of all three vehicles go in the same direction are

\(\displaystyle{A}={\left\lbrace{R}R{R},{L}{L}{L},{S}{S}{S}\right\rbrace}\).

Step 2

b): All possible outcomes of all three vehicles take different direction are

\(\displaystyle{B}={\left\lbrace{R}{L}{S},{R}{S}{L},{L}{R}{S},{L}{S}{R},{S}{R}{L},{S}{L}{R}\right\rbrace}\),

as we can see, the order of letters is important.

c): All possible outcomes of exactly two of the three vehicles turn right means that we need exactly two out of three letters to be "R"

\(\displaystyle{C}=\{R{R}{L},R{R}{S},{L}R{R},{S}R{R},{R}{S}{R},{R}{L}{R}\}\).

Step 3

d): All possible outcomes of exactly two vehicles go in the same direction is

\(\displaystyle{D}={\left\lbrace R{R}{L},R{R}{S},{L}R{R},{S}R{R},{R}{S}{R},{R}{L}{R},{L}{L}{R},{L}{L}{S},{R}{L}{L},{S}{L}{L},{L}{R}{L},{S}{S}{L},{S}{S}{R},{S}{R}{S},{S}{L}{S},{L}{S}{S},{R}{S}{S}\right\rbrace}\)

Here, every went should contain two of the same latter (any of the three) and the third letter should be different (any of the remaining two) from the one we took in the first step, this way we obtain that exactly two vehicles go in the same direction.

e): D' is the complement of event D. The complement (opposite) of D means that all vehicles go in the same direction or they go in different direction (to avoid that exactly two vehicles go in the same direction).

Therefore \(\displaystyle{D}'={\left\lbrace R{R}{R},{L}{L}{L},{S}{S}{S},{R}{L}{S},{R}{S}{L},{L}{R}{S},{L}{S}{R},{S}{R}{L},{S}{L}{R}\right\rbrace}\).

\(\displaystyle{C}\cup{D}\) is event C union event D. Event C is subset of event D (meaning that exactly two vehicles turn right is "subset" of exactly two vehicles go in the same direction). This indicates that \(\displaystyle{C}\cup{D}={D}\).

For the same reason we have that \(\displaystyle{C}\cap{D}\)(C intersection D) is

\(\displaystyle{C}\cap{D}={C}\).