Combinatorial interpretation of Fermat's Last Theorem

Combinatorial interpretation of Fermat's Last Theorem

A book that I am reading suggests the following exercise
Suppose we have $n > 2$ objects blue colored bins, red colored bins and
some uncolored bins. Show that Fermat's Last Theorem is equivalent to the
statement that
The number of ways of putting the objects into bins so that the bins of
both colors are shun is never equal to the number of ways to put the
objects into the bins so that neither color is shun.
The book also hints at defining $x$ to be the number of bins that are not
blue, $y$ the number of bins that are not red and $z$ the total number of
bins.
As far as I can come from here is that the above statement (if true) would
imply $(x+y-z)^n = z^n - x^n - y^n$ and I don't see how to proceed from
here.
Anyone happens to know how to prove the equivalence?