Question 19
In a school, every pair of students are either friends or strangers. A sequence of (not necessarily distinct) students $A_1, A_2, \dots, A_{2023}$ is called *mischievous* if
* Total number of friends of $A_1$ is odd.
* $A_i$ and $A_{i+1}$ are friends for $i = 1, 2, \dots, 2022$.
* Total number of friends of $A_{2023}$ is even.
Prove that the total number of *mischievous* sequences is even.
* Total number of friends of $A_1$ is odd.
* $A_i$ and $A_{i+1}$ are friends for $i = 1, 2, \dots, 2022$.
* Total number of friends of $A_{2023}$ is even.
Prove that the total number of *mischievous* sequences is even.