In the figure, we have a target at which darts are thrown. The inner circle is worth 10 points, the intermediate ring is worth 5 points, the outer ring is worth 3 points, and the external area is worth 0 points. If a dart hits a line, the score will be the average of the points of the regions divided by the line. In each turn, the player throws 4 darts.

![](https://cdn.mathpix.com/cropped/2024_05_01_3fc4f3a6ef7302a792fcg-03.jpg?height=507&width=513&top_left_y=1925&top_left_x=846)
a) The marks on the target in the following figure indicate the darts thrown by Leo. How many points did he score?

![](https://cdn.mathpix.com/cropped/2024_05_01_3fc4f3a6ef7302a792fcg-04.jpg?height=515&width=513&top_left_y=541&top_left_x=671)

b) How many different points can be obtained by throwing the 4 darts, without any of them hitting any line?

c) William intends to score 30 points in his 4 throws. In how many different ways can he achieve this sum, regardless of the sequence of the darts, i.e., the sequences $(10,10,10,0)$ and $(0,10,10,10)$ are considered the same way to sum 30 points $(3 \cdot 10=30)$?
Solution

a) Leo scored $10+5+\frac{5+3}{2}+\frac{3+0}{2}=20.5$ points.

b) The minimum score is 0 and the maximum is 40. Therefore, it is enough to verify for each of the natural numbers in this interval the possibility of being a possible sum. Let's construct a table, showing an example for each possible sum.

| $1^{\text{st}}$ dart | $2^{\text{nd}}$ dart | $3^{\text{rd}}$ dart | $4^{\text{th}}$ dart | Sum |
| :---: | :---: | :---: | :---: | :---: |
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 5 | 5 |
| 0 | 3 | 3 | 0 | 6 |
| 5 | 0 | 0 | 3 | 8 |
| 3 | 3 | 0 | 3 | 9 |
| 5 | 5 | 0 | 0 | 10 |
| 5 | 3 | 3 | 0 | 11 |
| 3 | 3 | 3 | 3 | 12 |
| 10 | 3 | 0 | 0 | 13 |
| 5 | 3 | 3 | 3 | 14 |
| 5 | 0 | 0 | 10 | 15 |
| 5 | 3 | 5 | 3 | 16 |
| 10 | 5 | 0 | 3 | 18 |
| 3 | 3 | 10 | 3 | 19 |
| 10 | 10 | 0 | 0 | 20 |
| 10 | 5 | 3 | 3 | 21 |
| 10 | 3 | 10 | 0 | 23 |
| 10 | 0 | 10 | 5 | 25 |
| 10 | 3 | 3 | 10 | 26 |
| 10 | 10 | 5 | 3 | 28 |
| 10 | 10 | 10 | 0 | 30 |
| 10 | 10 | 3 | 10 | 33 |
| 10 | 10 | 10 | 5 | 35 |
| 10 | 10 | 10 | 10 | 40 |
|  |  |  |  |  |

Therefore, there are 25 different scores.

c) In addition to the scores of 0, 3, 5, and 10, we have the scores of the lines which can be 1.5 or 4 or 7.5. To achieve 30 points without hitting the region worth 7.5, we must hit at least two darts in the region worth 10 points, that is, $(10; 10; 5; 5)$ or $(10; 10; 10; 0)$. In addition to these possibilities, we also have $(10; 7.5; 7.5; 5)$ and $(7.5; 7.5; 7.5; 7.5)$, totaling 4 possibilities altogether.