A Markov chain is defined as follows. There are $M$ coins $(M ∈ \mathbb N)$, each showing Heads or Tails. At each step, a coin is selected uniformly at random and flipped. The state of the system is the number of Heads.
(a) Determine the transition matrix of the Markov chain.
(b) Using any method, determine the stationary distribution (your solution should prove that the distribution you find is in fact the stationary distribution).
(c) Suppose $M = 20$ and the initial state is $X_0 = 10$. How long, on average, will it take until the next time the state is $10$?
My question
I am not sure for (b), how to prove its time reversibility.
My attempt so far
(a) $P_{i,i+1} = \frac{M-i}{2M}$, $P_{i,i-1} = \frac{i}{2M}$, $P_{i,i} = \frac{1}{2}$
(b) Using the property of time reversible Markov chain (without proving), I could get $\pi_0 = 2^{-M}, \pi_i = {M \choose i}2^{-M}$.
(c) I'm guessing that the average time to return to state $10$ is $\frac{1}{\pi_{10}}$. Still, I need to prove the time reversibility to be able to use it.