Question

Five sticks of length 1,3,5,7 and 9 feet are given. Three of these sticks and selected at random. What is the probability that selected sticks form a triangle.

Answer 1

​Given five sticks of lengths are 1,3,5,7 and 9 feet.
The possible lengths that form a triangle are (3,5,7) and (5, 7,9).
The number of favourable outcomes to form a triangle is $n\left(A\right)=2$.
and the total number of possible ways to select three sticks out of 5 sticks is $n\left(S\right){=}^5{C}_3=\frac{5!}{2!\times 3!}=\frac{5\times 4\times 3!}{2!\times 3!}=10$.
Therefore, the probability that selected sticks form a triangle $=\frac{n\left(A\right)}{n\left(S\right)}$
$=\frac{2}{10}$
$=\frac{1}{5}$