Problem Solving

If the operation * is defined for all integers a and b by a*b=a+b-ab,which of the foll must be true for all integers a,b and c??

I-a*b=b*a
II-a*0=a
III-(a*b)*c=a*(b*c)

I is true
a*b = a+b+ab
b*a = b+a+ba (ba= ab)

II is true
a*0 = a+0+0=a (any number multiplied by 0 is 0)

III is true
a*b = a+b+ab
(a*b)*c = a+b+ab+c+ca+cb+cab = a+b+c+ab+ac+bc+abc

b*c = b+c+bc
a*(b*c) = a+b+c+bc+ab+ac+abc = a+b+c+ab+ac+bc+abc

the answer would be the one that said I,II and III
which one is it???

You are right ...The answer is I II and III

But I have a question.The question says that a*b=a+b-ab..but you have mentioned in the answer as a*b=a+b+ab...Why is that??

sorry, my mistake, i read a+b+ab instead of a+b-ab
lucky for me the final result still holds true, in the exam this might not be true.

I is true
a*b = a+b-ab
b*a = b+a-ba (ba= ab)

II is true
a*0 = a+0-0=a (any number multiplied by 0 is 0)

III is true
a*b = a+b-ab
(a*b)*c = a+b-ab+c-c(a+b-ab) = a+b+c-ab-ac-bc+abc

b*c = b+c-bc
a*(b*c) = a+b-bc+c-a(b+c-bc) = a+b+c-ab-ac-bc+abc

Thank u