A bag contains 5 balls labeled with the numbers $3, 3, 4, 4, 4$, one each. Using this bag and one die, a trial is performed to obtain a score according to the following rule: If the ball drawn from the bag is labeled 3, roll the die 3 times and the sum of the three results is the score. If the ball drawn from the bag is labeled 4, roll the die 4 times and the sum of the four results is the score. What is the probability that the score obtained from one trial is 10 points? Express this as $\frac { q } { p }$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [1 point]
A bag contains 5 balls labeled with the numbers $3, 3, 4, 4, 4$, one each. Using this bag and one die, a trial is performed to obtain a score according to the following rule:
If the ball drawn from the bag is labeled 3, roll the die 3 times and the sum of the three results is the score.\\
If the ball drawn from the bag is labeled 4, roll the die 4 times and the sum of the four results is the score.
What is the probability that the score obtained from one trial is 10 points? Express this as $\frac { q } { p }$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [1 point]