If * be binary operation defined on $R$ by $a*b = 1 + ab,\forall a,b \in R$. Then, the operation * is

(i) commutative but not associative.

(ii) associative but not commutative.

(iii) neither commutative nor associative.

(iv) both commutative and associative.

(i) It is given that,, $a*b = 1 + ab,\forall a,b \in R$
$a*b = ab + 1 = b*a$

So, * is a commutative binary operation.

Also, $a*(b*c) = a*(1 + bc) = 1 + a(1 + bc)$

$a*(b*c) = 1 + a + abc$
…….(i)
$(a*b)*c = (1 + ab)*c$
$= 1 + (1 + ab)c = 1 + c + abc$

…..(ii)
From Eqs. (i) and (ii),
$a*(b*c) \ne (a*b)*c$

So, * is not associative
Hence we can say that, * is commutative but not associative.