The correct answer is c) past.
Past (preposition) - from one side to the other of something, or in front of something.
Rephrased: He passed his grandmother's house on his way to school.
Passed - from the verb to pass, past tense, coudln't be used in this sentence.
Passet is not a word.
Pasted - past form of the verb to paste, does not fit in the context.