Question

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
\[\left( a \right){{\text{ }}^{5}}{{C}_{3}}\]
\[\left( b \right){{\text{ }}^{5}}{{C}_{3}}-3\]
\[\left( c \right){{\text{ }}^{5}}{{C}_{3}}-2\]
\[\left( d \right){{\text{ }}^{5}}{{C}_{3}}-1\]

Answer 1

Hint: To solve the given question, we will first find out what a triangle is and how many sides it has. Then, we will make use of the fact that the sum of any two sides if a triangle is greater than the length of the third side. Thus, we will have to select the line segments in such a way that the sum of the lengths of any two line segments is greater than the length of the third line segment. For this, we will first find out the total number of ways in which we can select any three line segments, and then we will subtract those cases from it when the sum of the lengths of any two line segments is equal or smaller than the third side.

Complete step-by-step answer:
To start with, we must know that a triangle is a polygon having 3 sides. To solve this question, we will make use of the fact that, in a triangle, the sum of the lengths of any two sides is greater than the third side. Thus, if a, b and c are lengths of sides of a triangle, then they should follow the conditions.
\[a+b>c\]
\[b+c>a\]
\[a+c>b\]
Now, we have to select three line segments from the given line segments such that this condition is satisfied. For this, we will first find the total number of ways in which we can select 3 out of 5 lines. Total number of ways in which we can select r out of n distinct objects is \[^{n}{{C}_{r}}.\] Thus, the total number of ways in which we can select 3 out of 5 lines is \[^{5}{{C}_{3}}.\]
Now, we have to find those triplets of line segments when the condition of the triangle will not be satisfied. They are (2, 3, 5), (2, 3, 6) and (2, 4, 6). In these cases, we will not get a triangle. Thus, the required number of triangles will be obtained by subtracting these 3 cases from the total number of ways. Thus, we have,
\[\text{Number of triangles}={{\text{ }}^{5}}{{C}_{3}}-3\]
Hence, option (b) is the right answer.

Note: We can also solve the question alternately by finding all the triplets which will result in a triangle. These are (2, 3, 4), (3, 4, 5), (3, 4, 6), (2, 4, 5), (4, 5, 6), (3, 5, 6), (2, 5, 6). Thus, there are a total of 7 triplets. Therefore,
Number of triangles = 7
\[\Rightarrow \text{Number of triangles}=10-3\]
\[\Rightarrow \text{Number of triangles}={{\text{ }}^{5}}{{C}_{3}}-3\]