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Chapter 5: Problem 395

The straight lines \(\mathrm{I}_{1}, \mathrm{I}_{2}, \mathrm{I}_{3}\) are parallel and lie in the same plane. A total number of \(m\) points are taken on \(I_{1}, n\) points on \(\mathrm{I}_{2}, \mathrm{k}\) points on \(\mathrm{I}_{3}\). The maximum number of triangles formed with vertices at these points are (a) \(^{\mathrm{m}+\mathrm{n}+\mathrm{k}} \mathrm{C}_{3}\) (b) \({ }^{\mathrm{m}+\mathrm{n}+\overline{\mathrm{k}} \mathrm{C}_{3}-{ }^{\mathrm{m}}} \mathrm{C}_{3}-{ }^{\mathrm{n}} \mathrm{C}_{3}-{ }^{\mathrm{k}} \mathrm{C}_{3}\) (c) \({ }^{\mathrm{m}} \mathrm{C}_{3}+{ }^{\mathrm{n}} \mathrm{C}_{3}+{ }^{\mathrm{k}} \mathrm{C}_{3}\) (d) \(m+n+k-{ }^{m+n+k} C_{3}\)

Short Answer

Expert verified
The correct answer is \(m \cdot n \cdot k\).

Step by step solution

01

Total Points

Find the total number of points on all three lines as m + n + k.
02

Select Points on Each Line

Since we need to form a triangle, we need to pick one vertex on I1, one vertex on I2, and one vertex on I3. So, we will choose 1 point from m points, 1 point from n points, and 1 point from k points.
03

Calculate Combinations

Calculate the number of combinations possible by choosing 1 point from m, n, and k points using the combination formula. \(C_m = C^1_m = \frac{m!}{(m-1)!1!} = m \) \(C_n = C^1_n = \frac{n!}{(n-1)!1!} = n \) \(C_k = C^1_k = \frac{k!}{(k-1)!1!} = k \)
04

Calculate the Maximum Number of Triangles

Multiply the combinations to find the maximum number of triangles that can be formed. \(Triangularity = C_m \cdot C_n \cdot C_k = m \cdot n \cdot k \) Since this differs from all given options (a), (b), (c), and (d), it means that the problem statement is flawed or the given options are incorrect. The correct answer is \(m \cdot n \cdot k\) for the maximum number of triangles that can be formed with the given points on the three parallel lines.

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Most popular questions from this chapter

If \({ }^{189} \mathrm{C}_{35}+{ }^{189} \mathrm{C}_{\mathrm{x}}={ }^{190} \mathrm{C}_{\mathrm{x}}\) then \(\mathrm{x}\) is equal to (a) 34 (b) 35 (c) 36 (d) 37

If \(\left(1 /{ }^{4} \mathrm{C}_{\mathrm{n}}\right)=\left(1 /{ }^{5} \mathrm{C}_{\mathrm{n}}\right)+\left(1 /{ }^{6} \mathrm{C}_{\mathrm{n}}\right)\) then value of \(\mathrm{n}\) is (a) 3 (b) 4 (c) 0 (d) none of these

Number of points having position vector ai \(\wedge+b j \wedge+c k \wedge, a, b, c\) e \(\\{1,2,3,4,5\\}\) such that \(2^{a}+3^{b}+5^{c}\) is divisible by 4 is (a) 140 (b) 70 (c) 100 (d) 75

The number of 10 letter codes that can be formed using the characters \(\mathrm{x}, \mathrm{y}, \mathrm{z}, \mathrm{r}\) with the restriction that \(\mathrm{x}\) appears exactly thrice and \(\mathrm{y}\) appears exactly twice in each such codes is (a) 60840 (b) 88400 (c) 80640 (d) 64080

The maximum no. of points into which 4 circles and 4 straight lines intersect is (a) 26 (b) 56 (c) 50 (d) 72
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