Formal:
* Admittance to the museum is free for children under 12.
* The security guard checked our IDs for admittance to the building.
* The university's high admittance standards ensured a rigorous academic environment.
Informal:
* They refused her admittance to the party.
* You need a password for admittance to the online forum.
* The locked gate prevented admittance to the garden.
Figurative:
* His apology seemed to gain him admittance back into her good graces.
* The evidence presented in court opened the door to admittance of his guilt.
As you can see, "admittance" typically refers to the act of being allowed to enter a place or gain access to something. It can be used in both formal and informal contexts, and can also be used figuratively to describe gaining acceptance or access to something abstract.