The sentence with correct pronoun and antecedent agreement is:
Carol wanted to go outside, but it was raining and she could not find our rain jacket.
This sentence correctly uses "she" to refer to "Carol," which is singular and female.
The other sentences have issues with pronoun agreement:
- "Because the brothers were tired, he forgot..." incorrectly uses "he" to refer to "the brothers" (plural).
- "When the teachers arrived at school, she saw the students..." incorrectly uses "she" to refer to "the teachers" (plural).
The second sentence about the neighbors is also correct. It correctly uses "they" to refer to "the neighbors."
Thus, both the first and second sentences are correct, but since you asked for one, the first is a solid example.