admission
Permission to enter a place or join a group.
Admission means allowing someone to enter a place or become part of a group. When you buy admission to a museum or amusement park, you're paying for permission to go inside. Schools grant admission to new students, deciding who gets to attend based on applications and qualifications. A college might send you an admission letter saying you've been accepted.
The word also means acknowledging or confessing that something is true, especially something you'd rather not say. If you broke your sister's toy and make an admission of guilt, you're openly saying you did it. A scientist might make an admission that her earlier theory was wrong. This kind of admission takes courage because you're being honest about a mistake or uncomfortable truth.
Notice how both meanings involve a kind of opening: either opening a door to let someone in, or opening up to reveal something honest. When a detective gets a suspect to make an admission, the person is finally letting the truth come out. The related word admittance specifically means physical entry (No admittance beyond this point), while admission can mean entry or confession.