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

^{5} C_{3}

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^{5} C_{3} - 3

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^{5} C_{3} - 2

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^{5} C_{3} - 1

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Solution

The correct option is D $^{5} C_{3} - 3$
For a triangle we need $3$ segments and sum of any two segments must be greater than the other segment
Thus, $(2, 3, 5), (2, 3, 6)$ and $(2, 4, 6)$ have to be considered while making a selection and excluded.
No. of triangles $=^{5} C_{3} - 1 - 1 - 1$
$=^{5} C_{3} - 3$

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