Draw a triangle

We have line segments with lengths of 3cm, 5cm, 6cm, 7cm, and 9cm. What is the probability in % that if I randomly select three of them, I will be able to draw a triangle?

Correct answer:

p =  80 %

Step-by-step explanation:

$$\begin{gathered} C_3(5)=\left(\genfrac{}{}{0ex}{}{5}{3}\right)=\frac{5!}{3!(5-3)!}=\frac{5\cdot 4}{2\cdot 1}=10 \\ n=\left(\genfrac{}{}{0ex}{}{3}{5}\right)=10 \\ \Delta 3-5-6 \\ \Delta 3-5-7 \\ \Delta 3-6-7 \\ \Delta 3-7-9 \\ \Delta 5-6-7 \\ \Delta 5-6-9 \\ \Delta 5-7-9 \\ \Delta 6-7-9 \\ p=100\cdot \frac{8}{n}=100\cdot \frac{8}{10}=\frac{100\cdot 8}{10}=\frac{800}{10}=80\% \end{gathered}$$

Try another example