$.$ Using Boolean algebra, simplify the following expressions:

$($a$)($A $+$ B$)($A $+$ C$)($b$)$ AB $+$ ABC $+$ ABCD $+$ ABCDE

$($c$)$ AB $+$ ABC $+$ A $($d$)($A $+$ A$)($AB $+$ ABC$)$

$($e$)$ AB $+($A $+$ B$)$C $+$ A