Birthday wishes for friend
birthday wishes for a friend
(which one is correct)?
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To answer your question directly, "birthday wishes for a friend" is correct.
However, it depends on the context, as Mr. Ashworth noted in his comment. For instance, you could say,
"I'm writing birthday wishes for friends."
"I'm writing birthday wishes for a friend.
"Friend" is a countable noun, so it usually takes an article ("a" or "the") or a possessive (e.g., "my" or "your").
For more information about using articles before nouns in English, do a search on countable and uncountable nouns. Here is one useful link, but there are many more:
"[Countable] nouns refer to things that can be divided up into smaller units which are separate and distinct from one another. They usually refer to what can individually be seen or heard.
"[Uncountable] nouns refer to things that cannot be counted because they are regarded as wholes which cannot be divided into parts. They often refer to abstractions and occasionally have a collective meaning."
(Both from: http://leo.stcloudstate.edu/grammar/countnon.html)
In general, countable nouns can be pluralized by adding an "s" while uncountable nouns cannot be pluralized at all, although there are exceptions, including some nouns that can be both countable and uncountable depending on context.
"The lights hurt my eyes" but, "Light travels faster than sound."
Reading more about countable and uncountable nouns will improve your understanding of when to use an article before a noun.
Although only the version with an article is grammatical in standard English, the other version is acceptable as an instance of what's termed Headlinese - the style of writing used in newspaper headlines. One of the stylistic conventions of Headlinese is:
Articles are usually omitted. - wikipedia
Note that compressed telegraphic styles (or the modern equivalent, SMS language) are usually used within technological or spatial contexts that heavily favour very terse writing styles and penalise length. In these contexts, spelling and grammar rules are often bent or broken so long as there is some means for the receiver to recover the original meaning.