The correct option is "doesn't buy". The completed sentence is: "If he doesn't buy the tickets, they won't go to the concert."
Here's why:
The sentence describes a conditional situation. The first part of the sentence (the 'if' clause) uses the present simple tense to describe a condition that needs to be met. The second part uses 'will' + base form of the verb to show the likely result if the condition isn't met. Since "he" is a singular pronoun, it requires the third-person singular form of the verb "buy", which is "doesn't buy".
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