Consider two distinct positive numbers m, n with m > n. Let x = nlognm, y = mlogmn. The relation between x and y is-

The correct option is x > y.

Let us analyze the given expressions for $x$ and $y$ step-by-step.

We are given two distinct positive numbers $m$ and $n$ such that:
$m>n>0$

The expressions for $x$ and $y$ are:
$x=n^{{\log }_{n}m}$
and
$y=m^{{\log }_{m}n}$

Using the fundamental logarithmic identity, which states that for any positive base $b$ (where $b\ne 1$) and any positive number $a$:
$b^{{\log }_{b}a}=a$

We can simplify the expressions for $x$ and $y$ directly:

For $x$:
$x=n^{{\log }_{n}m}=m$

For $y$:
$y=m^{{\log }_{m}n}=n$

Since we are given that $m>n$, substituting the simplified values of $x$ and $y$ gives:
$x>y$

Thus, the relation between $x$ and $y$ is $x>y$.