Question

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

• A. ${}^6{C}_3-7$
• B. ${}^6{C}_3-1$
• C. ${}^6{C}_3$
• D. ${}^6{C}_3-2$

Answer 1

The correct option is A ${}^6{C}_3-7$

Given lengths of line segments are $2,3,4,5,6,7$,

The number ways of selecting $3$ lengths from them is ${}^6{C}_3$,

Consider the cases in which a triangle is not formed,

i.e., when any one of these two conditions is not satisfied,

$1)$sum of any two lengths is greater than the third length

$2)$difference between any two lengths is smaller than the third one,

Now,

number of such cases are

$2,3,5$

$2,3,6$

$2,3,7$

$2,4,6$

$2,4,7$

$2,5,7$

$3,4,7$

$\therefore$ the total number of ways are ${}^6{C}_3-7$