Question

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

Answer 1

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 $=^{6}C_{3}-6$
$=20-6$
$=14$