## Task 2 - 200732

Given are seven line segments with lengths $1 \mathrm{~cm}, 3 \mathrm{~cm}, 5 \mathrm{~cm}, 7 \mathrm{~cm}, 9 \mathrm{~cm}, 11 \mathrm{~cm}$, and $15 \mathrm{~cm}$.

a) Give the number of all different ways to select three of these seven line segments! Ways that differ only in the order of the selected line segments should not be considered as different.

b) Among the possibilities found in a), give all those for which a triangle can be constructed from the lengths of the three selected line segments as side lengths!

c) Calculate what percentage of the possibilities found in a) are the possibilities found in b)!

(The percentage should be rounded to one decimal place after the decimal point.)
The measures of the three selected line segments, measured in cm, are each called $a, b, c$. Due to the independence of their order, the following can hold:
$$
ac \quad \text { (1) } \quad b+c>a \quad(2) \quad c+a>b
$$

Because $ac ?$ | $a$ | $b$ | $c$ | Is $a+b>c$ true? | $a$ | $b$ | $c$ | Is $a+b>c$ true? | $a$ | $b$ | $c$ | Is $a+b>c$ true? |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 1 | 3 | 5 | No | 1 | 3 | 7 | No | 1 | 3 | 9 | No |
| 1 | 3 | 11 | No | 1 | 3 | 15 | No | 1 | 5 | 7 | No |
| 1 | 5 | 9 | No | 1 | 5 | 11 | No | 1 | 5 | 15 | No |
| 1 | 7 | 9 | No | 1 | 7 | 11 | No | 1 | 7 | 15 | No |
| 1 | 9 | 11 | No | 1 | 9 | 15 | No | 1 | 11 | 15 | No |
| 3 | 5 | 7 | Yes | 3 | 5 | 9 | No | 3 | 5 | 11 | No |


| $a$ | $b$ | $c$ | Is $a+b>c$ true? | $a$ | $b$ | $c$ | Is $a+b>c$ true? | $a$ | $b$ | $c$ | Is $a+b>c$ true? |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 3 | 5 | 15 | No | 3 | 7 | 9 | Yes | 3 | 7 | 11 | No |
| 3 | 7 | 15 | No | 3 | 9 | 11 | Yes | 3 | 9 | 15 | No |
| 3 | 11 | 15 | No | 5 | 7 | 9 | Yes | 5 | 7 | 11 | Yes |
| 5 | 7 | 15 | No | 5 | 9 | 11 | Yes | 5 | 9 | 15 | No |
| 5 | 11 | 15 | Yes | 7 | 9 | 11 | Yes | 7 | 9 | 15 | Yes |
| 7 | 11 | 15 | Yes | 9 | 11 | 15 | Yes |  |  |  |  |

From this, it follows:

The number sought in a) is 35, the number sought in b) is 11, the percentage sought in c) is $11 \cdot 10035 \% \approx 31.4 \%$.