4. Three people $A, B$ and $C$ play a game of passing a basketball from one to another. Find the number of ways of passing the ball starting with $A$ and reaching $A$ again on the 11 th pass. For example, one possible sequence of passing is
$$
A \rightarrow B \rightarrow A \rightarrow B \rightarrow C \rightarrow A \rightarrow B \rightarrow C \rightarrow B \rightarrow C \rightarrow B \rightarrow A .
$$
4. Ans: 682

Let $a_{k}$ denote the number of ways that the $k^{\text {th }}$ pass reach $A$. We have $a_{1}=0$. At each pass, the person holding the ball has 2 ways to pass the ball to. So total number of ways the ball can be passed after the $k^{\text {th }}$ pass is $2^{k}$ The number of ways that at the $(k+1)^{\text {th }}$ pass, $A$ receives the ball is $a_{k}+1$. So $a_{k+1}=2^{k}-a_{k}$ Thus $a_{1}=0, a_{2}=2$, $a_{3}=2, \ldots, a_{11}=682$.