The 277th National Junior High School Mathematics Competition consists of 14 questions (5 multiple-choice questions, 5 fill-in-the-blank questions, 4 problem-solving questions), with a full score of 150 points. Among them, each correct answer for multiple-choice and fill-in-the-blank questions earns 7 points, and a wrong answer earns 0 points, with no other point values; each problem-solving question is worth 20 points, and the step scores can only be $0, 5, 10, 15, 20$ points, with no other point values. How many different possible scores are there?
The score can be selected from 16 5s and 10 7s, let the total number of 5s and 7s selected be $k$.

Since selection and non-selection are relative, i.e., the score obtained by selecting $m$ 5s and $n$ 7s is the same as the score obtained by selecting $16-m$ 5s and $10-n$ 7s, which sums up to 150, we only need to consider half of the scores. Let the number of different scores that can be obtained between 0 and 74 be $a$. Since 75 can be obtained, the total number of different scores that can be obtained is $2a+1$. Therefore, we only need to discuss up to $k=14$.
(1) When $k=0$, the score is 0;
(2) When $k=2,4,6,8,10$, the scores are all even numbers between $10 \sim 14, 20 \sim 28, 30 \sim 42, 40 \sim 56, 50 \sim 70$ respectively;
(3) When $k=12,14$, the scores are all even numbers between $60 \sim 80, 70 \sim 90$ respectively;
(4) When $k=1$, the scores are all odd numbers between $5 \sim 7$;
(5) When $k=3,5,7,9$, the scores are all odd numbers between $15 \sim 21, 25 \sim 35, 35 \sim 49, 45 \sim 63$ respectively;
(6) When $k=11,13$, the scores are all odd numbers between $55 \sim 75, 65 \sim 85$ respectively.

In summary, the odd numbers that can be obtained between $0 \sim 74$ are 31, and the even numbers are 32. Therefore, $a=63$.
Thus, $2a+1=127$.