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Does the distribute of the survey happen when you click publish?
No, publishing only makes the survey accessible to those who have the link. If you want to send it to the entire university you can either use a contact list with everyone's email, or (if you do not want/need to collect information from a contact list) then you can send the link out via email (e.g., individual emails or a group email like [email protected])
you need to store emails and details for each member as contact list in directory and use this contact list to distribute survey via email.