The binary relation is defined on a set $A$ which is a function from $A*A$ to $A$ .
So , Number of element in $A*A$= $4*4=16$
So total number of binary operation is $4^{16}$.
It consist of all the binary relations but we need only commutative .
Commutative pair = no of symmetric relation which is first take all the reflexive pairs in $n$ ways then take symmetric pairs n(n-1)/2 .
So total =n+n(n-1)/2= n(n+1)/2.
putting n=4 ,we get 4*5/2=10 .
so total we have $4^{10}$ commutative relation.
