I think tye's solution is the best. If for some reason you would prefer one of the others, the answer would depend on how big n is compared to the size of the array. If it is small, say, smaller than 10%, then just pick a random number in the range 0..$#a and pick another one if that had been picked before. Such collisions should be rare for small n / @a; If it is larger, then shuffle first, but do not shuffle the array itself, especially if the array elements themselves tend to be big (which doesn't seem to be the case here). Instead, create a separate index array and shuffle that -- something like @indices = shuffle 0..$#a. Below, I tried to add comments to tye's code. I may have messed up, so wait until wiser monks have had a chance to peruse it.

use strict;
use warnings;

## How many elements to pick
use constant N => 10;

## The original array that must not be altered
my @letters = ('a'..'z');

##-----------------------------------------------------+
## Generate code that will pick an element from
##     @letters at each iteration without repeating
##-----------------------------------------------------+
my $picker_of_next_rand = gen_picker( \@letters );

## Now execute (iterate) that code N times
for (1..N) {
    print $picker_of_next_rand->();
}
print "\n";            ## look nice
print @letters, "\n";  ## original untouched
exit( 0 );

##-----------------------------------------------------+
## This sub generates and returns code that will pick
##     a random element from @$aref at each iteration
##     without repeating
##-----------------------------------------------------+
sub gen_picker {
    my ($aref) = @_;
    my @indices = 0 .. $#$aref;

    ## closes on @indices
    return sub {
        if (! @indices) {
            die "No more items left to pick from";
        }

        ##---------------------------------------------+
        ## pick an int in the range (0 .. $#$aref),
        ## the range of the array's indices
        ##---------------------------------------------+
        my $random_index = int rand @indices;

        ## $pick is the element at the random_index
        my $pick = $aref->[ $indices[ $random_index ] ];

        ##---------------------------------------------+
        ## We are about to pop the index array.  Save
        ## the element at the pop-end of the index array
        ## by copying it into the position of the $pick
        ## (clobber index of $pick, won't be used again)
        ##---------------------------------------------+
        $indices[ $random_index ] = $indices[ -1 ];

        ## Now it is safe to pop the index array
        pop @indices;

        return $pick;
    };
}
[download]

which prints something like the following:

vkdbfymxie
abcdefghijklmnopqrstuvwxyz
[download]

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