There are $5$ buckets and each of them can be opened and Closed i.e. $2$ options for each bucket.
so number of arrangements of buckets $= 2*2*2*2*2 = 2^5 = 32$
Now for of these $32$ arrangement a person can either select the arrangement or reject the arrangement i.e. $2$ options for each arrangement.
So the number of ways in which every person can select a set of arrangements such that set of his arrangements selected are different from other person's selection
$ = 2*2*2*2....32 \ times $
$= 2^{32} $
$= 2 ^{2^5} $
Hence Option $D.$ is the correct answer.