Design a DFA that recognizes the following language
As mentioned in the comments, you’ll need 8 states exactly. I like to label my states with semantic meanings, so in this case I’ll use a 3-digit number where each digit represents the number of as, bs, or cs we’ve seen ($\mod 2$). 

Then we have all the states from $000$ to $111$ in binary. 

You already know that $000$ is the start state and $010$ is the accept state. 

All that remains are the transitions, which are independent of the accept rules at this point. From each state, there are 3 transitions: one for each of a, b, and c.

But now notice that for each letter, the corresponding transition is to the state with the corresponding bit flipped! For example, $110 \xrightarrow{a} 010$.

More mathematically, for any state $\alpha\beta\gamma$:


when we see an a, we go to the state $\alpha'\beta\gamma$, where $\alpha'$ is 1 if $\alpha = 0$, and vice versa;
etc., for the remaining letters.