
Binary Operations!
Let S = {a, b, c, d}.
How many binary operations can be defined on S?
I think the answer might be $\displaystyle 4^{16}$
How many are commutative?
I think the answer is $\displaystyle 4^{10}$
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 $\displaystyle 4^{9}$
Can any1 tell me if my answers are right or wrong?

The answer really depends upon the definitions is use.
How are you defining binary operator?
The most common definition is a function $\displaystyle \phi :X \times X \mapsto X$.
If that is you definition then yes $\displaystyle 4^{16}$ is correct for the total number.
I agree with your other two answers.