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  • Find the number of triangles that may be created by 9 line segments of lengths 1, 2, 3, 4, 5, 6, 7, 8, and 9 units isOption: 1 63

Find the number of triangles that may be created by 9 line segments of lengths 1, 2, 3, 4, 5, 6, 7, 8, and 9 units is

Option: 1

63
 


Option: 2

72


Option: 3

57

 


Option: 4

69


Answers (1)

best_answer

Given that,

There are 9 line segments of lengths 1, 2, 3, 4, 5, 6, 7, 8, and 9.

The triangle is formed using 3 line segments.

We must select any three of the 9 line segments offered because a triangle has three line segments, and there are 9 of them.

Thus, the total number of ways to select 3 segments out of 9 is given by,

\mathrm{\begin{aligned} &{ }^9 C_3=\frac{9 !}{3 ! 6 !}\\ &{ }^9 C_3=\frac{9 \times 8 \times 7}{3 \times 2}\\ &{ }^9 C_3=84 \end{aligned}}

We know the property that in a triangle the sum of the two sides must be greater than the third side.

If we choose 1, 2, and 3 as the triangle's sides, then the third side is equal to the sum of the first two sides.

So,

1 + 2 = 3

It means if we take 1, 2, and 3 as the sides of the triangle then the triangle will not form.

Similarly, if we take sides 1, 3, and 4, the triangle will not form. 

Similarly, if we take sides 1, 4, and 5, the triangle will not form. 

Similarly, if we take sides 1, 5, and 6, the triangle will not form. 

Similarly, if we take sides 1, 6, and 7, the triangle will not form. 

Similarly, if we take sides 1, 7, and 8, the triangle will not form. 

Similarly, if we take sides 1, 8, and 9, the triangle will not form. 

Similarly, if we take sides 2, 3, and 5, the triangle will not form. 

Similarly, if we take sides 2, 4, and 6, the triangle will not form. 

Similarly, if we take sides 2, 5, and 7, the triangle will not form. 

Similarly, if we take sides 2, 6, and 8, the triangle will not form. 

Similarly, if we take sides 2, 7, and 9, the triangle will not form. 

Similarly, if we take sides 3, 4, and 7, the triangle will not form. 

Similarly, if we take sides 3, 5, and 8, the triangle will not form. 

Similarly, if we take sides 3, 6, and 9, the triangle will not form. 

Similarly, if we take sides 4, 5, and 9, the triangle will not form.

These side combinations are included in the overall number of possible ways to choose three line segments.

The entire number of possibilities to choose three line segments must therefore be subtracted from these 15 choices of sides.

Thus, the total number of triangles that can be formed by nine line segments is given by,

\mathrm{\begin{aligned} &{ }^9 C_3-15=84-15\\ &{ }^9 C_3-15=69 \end{aligned}}

Therefore, the total number of triangles formed is 69.

 

 

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Gunjita

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