How to proof that more than half binary algebraic operations on a finite set are non-commutative? We know that if $S$ is a set and $|S|=n$, then there are $n^{n^{2}}$ binary algebraic operations, right?
The cardinality of $|S|=n$ and cardinality of $|S\times S|=n^{2}$. Also, the number of all mirrors (or all functions) between these two sets will be $n^{n^{2}}$. We know that these functions are by other words: algebraic binary operations.  
But how do I to prove that there are $n^{\frac{n^{2}+n}{2}}$ commutative binary operations on $S$?
How do I prove that more than half binary algebraic operations on a finite set are non-commutative operations?

HINT: Let $S=\{s_1,s_2,\ldots,s_n\}$. If $\otimes$ is a commutative binary operation on $S$, then $\otimes$ is completely determined by the values of $s_j\otimes s_k$ for $1\le j\le k\le n$, and those values can be chosen independently of one another. 
