Question

How many triangles can be formed from the line segment of lengths $2,3,4,5,6,7?$

Answer 1

Answer: 14

Three line segments out of six can be chosen in
${}^6{C}_3^{}$ ways
For a triangle to be formed, the sum of lengths of two sides must be greater than the third side. Triangle is not formed for the triplets
$\left(2,3,5\right),\left(2,3,6\right),\left(2,3,7\right),\left(2,4,7\right),\left(2,5,7\right)\left(3,4,7\right)$
$\Rightarrow$ The required number ${=}^6{C}_3-6$
$=20-6$
$=14$