how do I simplify this particular boolean expression? so I have spent nearly 5 hours trying to simplify this particular expression but I keep going round and round in circles. I have my hard copy notes to show you where I scribbled for hours and hours to no end. so please can someone please show me how can this be done? 

$$(\overline{A}+\overline{B}+E)(\overline{A}+\overline{C}+D)(C+D+\overline{E})(\overline{B}+D)(A+E)\,.$$

I tried expansion, common, DeMorgan but I can't find my way out.
We can do it by brute force, expanding all 108 product terms and combining like terms carefully. Many of the terms can be eliminated immediately.

In hindsight or by luck or by sharp observation that D and E appear most frequently, we can proceed as follows.


  $(\overline{A}+\overline{B}+E)(\overline{A}+\overline{C}+D)(C+D+\overline{E})(\overline{B}+D)(A+E)$  


Let $D=1$ and $E = 1$. The formula becomes 11111 = 1
Let $D=1$ and $E = 0$. The formula becomes $(\overline{A}+\overline{B})1 1 1 A = \overline{B}A$
Let $D=0$ and $E =1$. The formula becomes $1(\overline{A}+\overline{C}) C \overline{B} 1 = \overline{A}C\overline{B}$
Let $D=0$ and $E = 0$. The formula becomes $(\overline{A}+\overline{B})(\overline{A}+\overline{C})1 \overline{B} A = \overline{C}\overline{B}A$

So the original formula is
$$DE + (D\overline{E}A + \overline{D}(E\overline{A}C + \overline{E}\overline{C}A))\overline{B}$$

(Computationally, it is debatable which formula is simpler.)