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Both expressions are correct, but they are used slightly differently. "Alles Gute zum Geburtstag" is more casual and can be used among friends and acquaintances. It’s a warm, friendly way to say happy birthday in German without sounding too formal. On the other hand, "Herzlichen Glückwunsch zum Geburtstag" is the full formal version and is ideal for professional settings, older people, or people you don’t know very well. If you're writing a card or email for someone at work, go with the formal version. If it’s your sibling or best friend, the simpler one works well. Ultimately, both phrases express the sentiment of happy birthday in German, and the choice depends on the relationship and setting.

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