4. Ali plays a trivia game with 5 categories, each with 3 questions. She earns 1 point for each correct answer. If she answers all 3 questions in a category correctly, she earns 1 bonus point. Ali answers exactly 12 questions correctly and the remaining 3 questions incorrectly. What are her possible total scores?
4. Ali earns a total of 12 points for her 12 correct answers. To determine her possible total scores, we need to determine her possible numbers of bonus points.
Since Ali answers 3 questions incorrectly, these could all be in 1 category, split over 2 categories (2 from 1 category and 1 from another), or split over 3 categories ( 1 from each).
In the first case, she answers all of the questions in 4 of the 5 categories correctly, and so earns 4 bonus points. In this case, her total score would be $12+4=16$.
In the second case, she answers all of the questions in 3 of the 5 categories correctly, and so earns 3 bonus points. In this case, her total score would be $12+3=15$.
In the third case, she answers all of the questions in 2 of the 5 categories correctly, and so earns 2 bonus points. In this case, her total score would be $12+2=14$.
These are all of the possibilities.
Therefore, Ali's possible total scores are 14,15 and 16 .
Answer: $14,15,16$