This language is recognized by the <parity-counting deterministic finite automaton> with four states
$$
Q=\{(p_b,p_c):p_b,p_c\in\mathbb Z_2\}.
$$
The initial state is $(0,0)$ and the only accepting state is $(1,0)$. Reading $a$ leaves the state unchanged, reading $b$ toggles $p_b$, and reading $c$ toggles $p_c$. The state therefore records exactly the parities of the numbers of $b$'s and $c$'s. Thus
$$
\boxed{\text{the language is regular}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 4H / a / ii / Solution

To recognize prescribed parities of several symbol counts, use one state for each tuple of parities. Reading a tracked symbol toggles its coordinate and leaves all other coordinates unchanged. For two tracked symbols this gives four states indexed by $\mathbb Z_2^2$.


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