Simple Combinatorics

First of all , to form a triangle there must be 3 line segments .Therefore choose 3 line segments as $6 \choose 3$ . But among these groups, there are 7 groups which do not satisfy the triangle inequality : For a $\Delta$ ABC : For the triangle to exist , sum of any two sides must be greater than the third .

The groups are : (2,3,7) , (2,3,6) , (2,3,5) , (2,4,6) , (2,4,7) , (2,5,7) and (3,4,7) .

Therefore the required answer is : $6 \choose 3$ - $7$ = $\boxed{13}$

I used facilities from the Python sympy and itertools modules. (Programmers are supposed to be lazy.)

The code attempts to construct a triangle from each combination of three lengths. If the result is indeed a triangle then the code increments its count.

Note that in triangle ABC one segment Plus another segment bigger than the other else segment.so we must choose 3 numbers which the sum of the two of them ‏ are bigger than another segment. (2.3.4) (2.4.5) (2.5.6) (2.6.7) Those are for two (3.4.5) ( 3.4.6) (3.4.7) (3.5.6) ( 3.5.7)
(3.6.7) Those are for three (4.5.6) ( 4.5.7) (4.6.7) And these are for four So we have 13 triangles