
der Eintritt
admission fee
The money you have to pay in order to enter a place (e.g. museum, cinema, swimming pool) or take part in an event.
Der Eintritt für das Konzert kostet 25 Euro.
Admission to the concert costs 25 euros.
admission / entry
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admission fee
The money you have to pay in order to enter a place (e.g. museum, cinema, swimming pool) or take part in an event.
Der Eintritt für das Konzert kostet 25 Euro.
Admission to the concert costs 25 euros.

entry / joining
The act of entering a room or becoming a member of an organization.
Nach seinem Eintritt in die Partei übernahm er schnell Verantwortung.
After joining the party, he quickly took on responsibility.

onset / beginning
The beginning of a state, period of time, or event.
Mit Eintritt der Dunkelheit wurde es merklich kühler.
With the onset of darkness, it became noticeably cooler.
The noun 'der Eintritt' (admission/entry) is directly derived from the separable verb 'eintreten' (to enter), which literally means 'to step in'. So 'Eintritt' is the act of going in or the result of it.
Bitte warten Sie nach dem Eintreten im Foyer.
Please wait in the foyer after entering.
When 'Eintritt' means the admission fee, it is often used with verbs such as 'kosten' (to cost), 'bezahlen' (to pay), or 'verlangen' (to charge/demand). A common and useful expression is also 'freier Eintritt' (free admission).
Wie viel kostet der Eintritt ins Museum?
How much does admission to the museum cost?
In the meaning of joining, 'Eintritt' is almost always used with the preposition 'in' followed by an accusative object. This describes which organization or group one is joining.
Sein Eintritt in den Sportverein war eine gute Entscheidung.
His joining the sports club was a good decision.
The use of 'Eintritt' for the beginning of an event, as in 'Eintritt der Dunkelheit' (onset of darkness), is typical of written language or a formal style. It often sounds poetic or official.
Bei Eintritt des Frühlings erwacht die Natur.
With the onset of spring, nature awakens.
beginning
Beginn generally refers to the temporal beginning of an event, an action, or a period of time. Eintritt is used especially when a state, an effect, or a period of time begins. With states, both are possible, but Eintritt emphasizes the change more strongly.

Mit Eintritt der Dunkelheit gingen die Wanderer nach Hause.
At the onset of darkness, the hikers went home.
Zu Beginn der Dunkelheit gingen die Wanderer nach Hause.
As darkness began to fall, the hikers went home.