This does not consider ajax, but it might be a help
<select name="cars">
@foreach($carTypesList as $type =>$name)
<option value="{{ $type }}"
@if(old('cars',$profile->car) == {{ $type }})
selected
@endif
>{{ $name }}
</option>
@endforeach
</select>
if you don't like mixing in the test in with the html so much, you could duplicate the line like;
<select name="cars">
@foreach($carTypesList as $type =>$name)
@if(old('cars',$profile->car) == $type )
<option value="{{ $type }}" selected >{{ $name }}</option>
@else
<option value="{{ $type }}">{{ $name }}</option>
@endif
@endforeach
</select>
Whilst this is not 'dry' it's easier to read and therefore quicker to work with.
This is from a post I contributed to a few weeks ago...https://laracasts.com/discuss/channels/laravel/old-data-didnt-display-in-edit-page