= Solution
Let $T_i$ be the expected time to complete the specified directed transition, starting at $i$. This is a <transition-completion time in a Markov chain>, rather than merely a first visit to $c$. First-step decomposition gives
$$
T_a=1+\tfrac35T_b+\tfrac25T_c,\qquad
T_b=1+\tfrac34T_a,\qquad
T_c=1+\tfrac23T_a+\tfrac13T_b.
$$
The missing $b\to c$ continuation in the middle equation represents successful completion, but its step is counted by the leading one. Substitution gives
$$
\boxed{T_a=128/11,\qquad T_b=107/11,\qquad T_c=12.}
$$
Thus the requested expectation from $a$ is $128/11$ steps.
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