Simplify a logic expression - Where have I gone wrong?
Both answers are correct. 

Let:
$$
f_1(A,B,C) = AB+AC+\overline{C}\\
f_2(A,B,C) = A+\overline{C}
$$
Let's build the thruth table:
$$\begin{array}{|c|c|c|c|c|}
\hline
A &amp; B &amp; C &amp; f_1 &amp; f_2 \\
\hline
0 &amp; 0 &amp; 0 &amp; 1 &amp; 1\\
\hline
0 &amp; 0 &amp; 1 &amp; 0 &amp; 0\\
\hline
0 &amp; 1 &amp; 0 &amp; 1 &amp; 1\\
\hline
0 &amp; 1 &amp; 1 &amp; 0 &amp; 0\\
\hline
1 &amp; 0 &amp; 0 &amp; 1 &amp; 1\\
\hline
1 &amp; 0 &amp; 1 &amp; 1 &amp; 1\\
\hline
1 &amp; 1 &amp; 0 &amp; 1 &amp; 1\\
\hline
1 &amp; 1 &amp; 1 &amp; 1 &amp; 1\\
\hline
\end{array}$$

As you can see the two functions correspond.

Please note that this tabular method of proving that two functions are the same is perfectly valid and is called Proof by exhaustion.