Correct answer # 2: The world's nations must cooperate; one nation cannot solve the world's problems by itself.
When you connect two related independent clauses or sentences you use a semicolon. Both sentences should be completed grammarly writtend and with a connection. Make sure you do not write a capital letter after the semicolon, unless it is a proper name.
Answer # 3 is incorrect as there is a capitalized word
Answer # 1 it is incorrect to use "for" as the sentence is grammarly completed