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Permutations and Combinations question

2024 · 6 Apr · Shift 1 · Q47
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Permutations and Combinations question

2024 · 6 Apr · Shift 1 · Q47

JEE MainMathematicsPermutations and CombinationsMCQ+4 / −1
The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is
  1. A
    56
  2. B
    16
  3. C
    24
  4. D
    48
View written solutionFree

Correct answer: B

  1. Total triangles from 8 vertices

A regular octagon has 888 vertices. Any choice of 333 vertices forms a triangle.

Hence total number of triangles is

(83)=56.\binom{8}{3} = 56.(38​)=56.
  1. Condition: no side of the triangle should be a side of the octagon

A side of the triangle will coincide with a side of the octagon exactly when two chosen vertices are adjacent vertices of the octagon.

So we need the number of ways to choose 333 vertices from 888 such that no two chosen vertices are adjacent.

  1. Count selections of 3 non-adjacent vertices on a circle of 8 vertices

We count the number of ways to choose 333 vertices from 888 arranged in a circle, with no two consecutive.

For a circular arrangement, the number of ways to choose rrr non-consecutive objects from nnn objects is

nn−r(n−rr).\frac{n}{n-r}\binom{n-r}{r}.n−rn​(rn−r​).

Here n=8n=8n=8, r=3r=3r=3. Thus

88−3(8−33)=85(53)=85⋅10=16.\frac{8}{8-3}\binom{8-3}{3} = \frac{8}{5}\binom{5}{3} = \frac{8}{5}\cdot 10 = 16.8−38​(38−3​)=58​(35​)=58​⋅10=16.
  1. Therefore

The required number of triangles is

16.16.16.
  1. Check options
  • A: 565656 — total triangles, does not satisfy restriction
  • B: 161616 — correct
  • C: 242424 — incorrect
  • D: 484848 — incorrect

So the correct option is B.

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