Question

Given five line segments of length 2, 3, 4, 5, 6 units. Then the number of triangles that can be formed by joining these lines is

• A. ${}^5{C}_3-3$
• B. ${}^5{C}_3-1$
• C. ${}^5{C}_3$
• D. ${}^5{C}_3-2$

Answer 1

The correct option is A ${}^5{C}_3-3$

First choose any 3 line segments out of 5 to make a triangle which can be done in ${}^5{C}_3$ ways.
As we know that sum of any two edge length should be greater than third one in triangle. So, rule out some combinations of line segment.
(2,3,5),(2,3,6) & (2,4,6)
Hence, total number of triangle ${=}^5{C}_3-3$