Simple Boolean Algebra Question

Question

I have the following term in front of me:

$$(AB+AC+\overline BC+B\overline C)*(A+\overline B+C+D)$$

and just need to multiply the whole thing which should result in this:

$$(AB+ABC+ABD+AC+A\overline BC+ACD+\overline BC+\overline BCD+AB\overline C+B\overline CD)$$

I just don't get why I have AB, AC and $\overline BC$ again but not $B\overline C$.

Stupid question but I am missing this basic stuff.

Answer 1

You have $AB$ again because it equals $(AB)(A)$, which is a product of one term from the first factor and one term from the second. You have $AC$ again because it equals $(AC)(A)$ (and also $(AC)(C)$), which is a product of one term from the first factor and one term from the second. You have $\bar BC$ again because it equals $(\bar BC)(\bar B)$ (and also $(\bar BC)(C)$), which is a product of one term from the first factor and one term from the second. But $B\bar C$ is not such a product.