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How many binary relations are there on a set $A$ with $n$ elements?

Total number of  binary relation from n element set to itself is $2^{n^{2}}$ I.e.n2 entries with two choice take it or not.

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∣A∣ =n

∣A⨉A∣ = n^2

Relation is the subsets of A⨉A, so the total no of binary relation on set A = cardinality of power set of (A⨉A)  = ∣ p(A⨉A) ∣ =2^(n^2)

So, the correct answer is 2^(n^2).