Factorize each of the following expression:

$a b-a-b+1=(a b-b)+(1-a)$ [Regrouping the expressions]

$=b(a-1)+(1-a)$

$=b(a-1)-(a-1)$      $[\because(1-a)=-(a-1)]$

$=(a-1)(b-1)$          [Taking out the common factor $(a-1)$ ]