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) $^{6}{{C}_{3}}-8$
(B) $^{6}{{C}_{3}}-6$
(C) $^{6}{{C}_{3}}-5$
(D) $^{6}{{C}_{3}}-4$

Answer 1

Hint: We know that in a triangle there are three line segments and we have six line segments. So, we have to choose three line segments. We know the formula, the total number of ways to select m items out of n given items, $^{n}{{C}_{m}}$ . Use this formula and get the total number of ways to select three line segments out of three six-line segments. We know the property that in a triangle the sum of the two sides must be greater than the third side. Use this property and get those combinations of sides that cannot form a triangle. Now, deduct the number of these combinations from the total number of ways to select three line segments out of six line segments.

Complete step-by-step answer:
According to the question, we have six line segments of lengths 2 units, 3 units, 4 units, 5 units, 6 units, and 7 units.
We know that a triangle has 3 line segments.
Since a triangle has 3 line segments and we are given six line segments, so we have to choose any three line segments out of six line segments given.
We know the formula, the total number of ways to select m items out of n given items, $^{n}{{C}_{m}}$ ……………………………(1)
We have to choose any three line segments out of six given line segments.
Now, using the formula shown in equation (1), we get
The total number of ways to select 3 line segments out of six segments = $^{6}{{C}_{3}}$ ……………………………..(2)
We know the property that in a triangle the sum of the two sides must be greater than the third side.
If we select 2, 3, and 5 as the sides of the triangle, then the summation of two sides is equal to the third side. That is,
$2+3=5$ .
It means if we take 2, 3, and 5 as the sides of the triangle then triangle will not form ………………….(3)
Similarly, the triangle will not form if we take the sides 2, 4, and 6 …………………………………….(4)
Similarly, the triangle will not form if we take the sides 2, 5, and 7 ………………………………..(5)
Similarly, the triangle will not form if we take the sides 3, 4, and 7 ……………………………………….(6)
From equation (3), equation (4), equation (5), and equation (6), we have those combinations of sides which cannot form a triangle.
The total number of ways to select three line segments also includes these combinations of sides.
So, we have to deduct these 4 combinations of sides from the total number of ways to select three line segments.
Subtracting, we get
$^{6}{{C}_{3}}-4$ ……………………………(8)
Therefore, the total number of triangles that can be formed by six line segments of length 2, 3, 4, 5, 6, 7 units is $^{6}{{C}_{3}}-4$ .
Hence, option (D) is the correct one.

Note: In this question, one might think that a triangle has three line segments, so we have to select any three line segments out of six line segments. But every combination of three sides doesn’t form a triangle. Some combinations of three sides do not form a triangle because we have a property that the sum of two sides must always be greater than the third one. Due to this the combination of the lines segments $\left( 2,3,5 \right)$ , $\left( 2,4,6 \right)$ , $\left( 2,5,7 \right)$ , and $\left( 3,4,7 \right)$ doesn’t form a triangle. So, we need to deduct these four combinations of sides from the total number of ways to select three line segments out of sic line segments, $^{6}{{C}_{3}}-4$ .