How can I simplify the following Product of Sums: ab+ac'+a'b'+a'c+b'c'+bc?
Update It turns out that your initial simplification was incorrect. Here's a correct one. First, observe that 
$$
(p+q)(p+q') = pp+pq'+pq+qq'=p +p(q'+q)+0 = p+p1=p
$$
so for the first two terms in your POS we have
$$
((a+c)+b)((a+c)+b') = a+c
$$
and, similarly, for the last two terms we have
$$
((a'+b)+c)((a'+b)+c') = a'+b
$$
so the full product simplifies to
$$
(a+c)(a'+b)\quad\text{ or, if you prefer }\quad ab+a'c+bc
$$
At this stage you haven't simplified this as much as possible. If you construct truth tables, you'll discover that $ab+a'c+bc=ab+a'c$, in other words the $bc$ term isn't needed here. It isn't obvious how one would discover that unless you've seen Karnaugh maps, for instance.