The answer really depends upon the definitions is use.
How are you defining binary operator?
The most common definition is a function .
If that is you definition then yes is correct for the total number.
I agree with your other two answers.
Let S = {a, b, c, d}.
How many binary operations can be defined on S?
I think the answer might be
How many are commutative?
I think the answer is
Assume that x = x * a = a * x for all x in the set S.
Then how many binary operations are there?
I think the answer is
Can any1 tell me if my answers are right or wrong?