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Chapter 5: Problem 405

There are 10 points in a plane out of these 6 are collinear. If \(\mathrm{N}\) is the number of triangles formed by joining these points then: (a) \(\mathrm{N} \leq 100\) (b) \(100<\mathrm{N}<140\) (c) \(140<\mathrm{N}<190\) (d) \(\mathrm{N}>190\)

Short Answer

Expert verified
The number of triangles that can be formed by joining these points is \( \mathrm{N} \). We found that \( \mathrm{N} = 100 \), so the correct answer is (a) \( \mathrm{N \leq 100} \).

Step by step solution

01

Determine the number of triangles formed by all 10 points

First, let's find out how many triangles can be formed by all 10 points in a plane without any restriction. We can choose any 3 points out of 10, and the number of ways this can be done is given by the combination formula: \( \mathrm{C(n, k) = \frac{n!}{k!(n-k)!} \) Here, n = 10 (total points), and k = 3 (points needed to form a triangle). So: \( \mathrm{C(10, 3) = \frac{10!}{3! \cdot (10-3)!} }\)
02

Calculate C(10, 3)

Now, let's calculate the C(10, 3) value: \( \mathrm{C(10, 3) = \frac{10!}{3! \cdot 7!} = \frac{10 \cdot 9 \cdot 8}{3 \cdot 2 \cdot 1} = 120 }\) So, there are 120 ways to form triangles using all 10 points.
03

Determine the number of triangles formed by collinear points

Since 6 points are collinear, they all lie on a straight line. We need to determine the number of triangles that can be formed by choosing 3 points out of the 6 collinear points. Again, we will use the combination formula with n = 6 and k = 3: \( \mathrm{C(6, 3) = \frac{6!}{3! \cdot (6-3)!} }\)
04

Calculate C(6, 3)

Now, let's calculate the C(6, 3) value: \( \mathrm{C(6, 3) = \frac{6!}{3! \cdot 3!} = \frac{6 \cdot 5 \cdot 4}{3 \cdot 2 \cdot 1} = 20 }\) So, there are 20 ways to form triangles using the 6 collinear points. However, since these points are collinear, the triangles formed will also be collinear, and thus, they will not be valid triangles.
05

Find the valid number of triangles

Finally, we need to subtract the number of collinear triangles from the total number of triangles formed using all 10 points: \( \mathrm{N = 120 - 20 = 100} \) Now, we can compare the value of N to the options given in the problem: (a) \( \mathrm{N \leq 100} \) (b) \( 100<\mathrm{N}<140 \) (c) \( 140<\mathrm{N}<190 \) (d) \( \mathrm{N}>190 \) As N = 100, the correct answer is (a) \( \mathrm{N \leq 100} \).

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