Final answer:
The statement is false. MAPI and ActiveSync can be used to connect on-premise Exchange servers with Office 365, allowing synchronization across environments.
Step-by-step explanation:
The statement that MAPI and ActiveSync do not permit the connecting of on-premise mailboxes to Office 365 mailboxes is false. MAPI (Messaging Application Programming Interface) is a protocol that allows Microsoft Outlook to communicate with an email server, while ActiveSync is a synchronization protocol designed to synchronize emails, contacts, calendars, tasks, and notes from a messaging server to mobile devices. Both protocols can be used in hybrid environments to connect on-premise Exchange servers with Office 365.