A total of $m$ male and $m$ female students are distributed into two classrooms, with $m$ students per classroom. At each time, a student is randomly selected from each classroom and the two students are interchanged. Let $X_n$ denote the number of female students in classroom 1 at time $n$, with states $\{0, 1, ..., m \}$. Then the process $\{X_n; n = 0, 1, ... \}$ is a Markov chain.
Letting $\pi_i$ denote the stationary probability of state $i$, show that it is time reversible.
Not much of an attempt, but I tried this: I will denote $P_{ij}$ as the probability of going from state $i$ to $j$.
Then $\pi_0 = P_{10}\pi_1$
$\pi_1 = P_{01}\pi_0 + P_{11}\pi_1 + P_{21}\pi_2 = \pi_0 + P_{11}\pi_1 + P_{21}\pi_2$
...
$\pi_j = P_{j-1,j}\pi_{j-1} + P_{j,j}\pi_j + P_{j+1,j}\pi_{j+1}$
But I do not know how to get $P_{j,i}\pi_j = P_{i,j}\pi_i$