You will often find a line like this in `/etc/sudoers`:

    # Allow members of group sudo to execute any command
    %wheel   ALL=(ALL:ALL) ALL

This will allow any user that is in the "_wheel_" group to make use of `sudo` with suitable proof of identity (e.g: their password). The nominated group may also be "_sudo_", "_admin_", or others... (e.g: line starts with `%sudo`)

If this is present in the file, then run `id` to see what groups you're in:

    $ id
    uid=1000(attie) gid=1000(attie) groups=1000(attie),27(sudo),117(docker)

If your user isn't in the appropriate group, then you must add your user to that group.

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An alternative would be to list both of your rules one-by-one, with the last matching rule taking effect (i.e: order is important):

    username ALL=(ALL) ALL
    username ALL=(ALL) NOPASSWD: /home/username/script.sh

See the ArchWiki page on sudo: https://wiki.archlinux.org/index.php/sudo#Example_entries