Question

Given six line segments of length $2,3,4,5,6,7$units, the number of triangles that can be formed by these lines is?

• A. ${}^{6}C_3-7$
• B. ${}^{6}C_3-6$
• C. ${}^{6}C_3-5$
• D. ${}^{6}C_3-4$

Answer 1

The correct option is A

${}^{6}C_3-7$

To find the number of triangles formed:

Given six line segments of length $2,3,4,5,6,7$units.

We need three line segments to form a triangle that can be selected in ${}^{6}C_3$ways.

Now, as per the geometry, there are two cases that satisfy the conditions for triangles.

$\left(i\right)$ The sum of any two sides of a triangle is greater than the third side.

$\left(ii\right)$ The difference between any two sides of a triangle is less than the third side.

Thus, based on these two conditions, the combination of three line segments that do not satisfy the triangle are:

$(2,3,5),(2,3,6),(2,3,7),(2,4,6),(2,4,7),(2,5,7)$ and $(3,4,7)$

Thus, there are a total of $7$such cases.

Therefore, the number of triangles formed here $={}^{6}C_3-7$

Hence, the correct option is A.