9. Five friends must get off a five-seat chairlift and can do so by going in three different directions: left, straight, or right. Getting off a chairlift, it is easy to collide with your fellow riders. For example: if I decide to go straight and someone to my left decides to go right, we collide; the same happens if I decide to go right and someone to my right goes straight (or left); if, however, someone goes in the same direction as me, we do not collide; and so on. If each of the five friends chooses at random where to go, with a probability of $1 / 3$ for each direction, what is the probability that there are no collisions?
(A) $\frac{25}{27}$
(B) $\frac{3}{5}$
(C) $\frac{40}{81}$
(D) $\frac{1}{3}$
(E) $\frac{7}{81}$
9. The answer is $\mathbf{( E )}$. Let's start by counting the total number of cases, i.e., the strings of length 5 composed of the letters S, A, and D (Left, Forward, and Right). Each of the five friends can independently choose one of the three directions, so these strings are $3^{5}=243$.

Now let's think about which strings result in no accidents: they are the strings where all the $\mathrm{S}$ (even 0) appear before all the A (even 0), which appear before all the $\mathrm{D}$ (even 0). We need to find all the strings of five letters that satisfy this condition. In other words, we need to write 5 as the sum of 3 ordered non-negative addends. This can be done in $\left(\begin{array}{l}7 \\ 2\end{array}\right)=21$ ways. In conclusion, the probability that there are no collisions is $7 / 81$.