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Q.

Given six-line segments of length2, 3, 4, 5, 6, 7units. Then the number of triangles that can be formed by joining these lines is:

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a

^{6} C_{3} - 7

b

^{6} C_{3} - 1

c

^{6} C_{3}

d

^{6} C_{3} - 2

answer is A.

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Detailed Solution

Given,
The six line segments with lengths = 2, 3, 4, 5, 6, 7,
Total 3 line segments needed to make a triangle which can be picked out

=^{6} C_{3},

ways.
There are now two ways by which requirements for triangles does not meet.
1) Any two lengths which are added together are longer than the third length.
2) Difference of any two length are less than the 3rd one,
So, we have total 7 such points which do not satisfy the required conditions,

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

And

(3, 4, 7)

Thus, the total number of required ways

=^{6} C_{3} - 7

Hence, option 1 is Correct.

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