There is a bag containing 5 balls with the numbers $3,3,4,4,4$ written on them, one each. Using this bag and one die, a trial is performed to obtain a score according to the following rule. A ball is randomly drawn from the bag. If the number on the drawn ball is 3, the die is rolled 3 times and the sum of the three numbers shown is the score. If the number on the drawn ball is 4, the die is rolled 4 times and the sum of the four numbers shown is the score. What is the probability that the score obtained from one trial is 10 points? [4 points] (1) $\frac { 13 } { 180 }$ (2) $\frac { 41 } { 540 }$ (3) $\frac { 43 } { 540 }$ (4) $\frac { 1 } { 12 }$ (5) $\frac { 47 } { 540 }$
There is a bag containing 5 balls with the numbers $3,3,4,4,4$ written on them, one each. Using this bag and one die, a trial is performed to obtain a score according to the following rule.
A ball is randomly drawn from the bag. If the number on the drawn ball is 3, the die is rolled 3 times and the sum of the three numbers shown is the score.\\
If the number on the drawn ball is 4, the die is rolled 4 times and the sum of the four numbers shown is the score.
What is the probability that the score obtained from one trial is 10 points? [4 points]\\
(1) $\frac { 13 } { 180 }$\\
(2) $\frac { 41 } { 540 }$\\
(3) $\frac { 43 } { 540 }$\\
(4) $\frac { 1 } { 12 }$\\
(5) $\frac { 47 } { 540 }$