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Of the 8 points in a given plane 4 are on circle O -Geometry - PS

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Of the 8 points in a given plane 4 are on circle O -Geometry - PS  [#permalink]

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New post 24 Aug 2018, 20:00
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Of the 8 points in a given plane 4 are on circle O and remaining 4 are on line L. Number of different quadrilaterals that can be formed by these 8 points are

A) 52

B) 53

C) 54

D) 64

E) 74

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Re: Of the 8 points in a given plane 4 are on circle O -Geometry - PS  [#permalink]

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New post 24 Aug 2018, 21:02
4C3*4C1 + 4C4 + 4C2*4C2 = 53

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Of the 8 points in a given plane 4 are on circle O -Geometry - PS  [#permalink]

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New post 25 Aug 2018, 00:29
Attachment:
Line_Circle.png


We are given a line and circle each which contain 4 points. The 8 points can be the vertices of a quadrilateral.
A line which has 4 collinear points can have a maximum of 2 points which can be the vertices. There are 3 ways
in which a quadrilateral can be formed

1. All 4 vertices are on the circle - \(C_4^4 = 1\)
2. Three vertices are on the circle and one vertice if on the line - \(C_3^4*C_1^4 = 4*4 = 16\)
3. Two vertices are on the circle and two vertices are on the line - \(C_2^4*C_2^4 = 6*6 = 36\)

Therefore, the total number of quadrilaterals that can be formed by these points are 1+16+36 = 53(Option B)
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Re: Of the 8 points in a given plane 4 are on circle O -Geometry - PS  [#permalink]

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New post 25 Aug 2018, 01:30
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Out of the four points given, A quadrilateral can be formed by selecting \(4\) points from the \(8\) points except when:
i) \(4\) points have been taken from the line
ii) \(3\) points have been taken from the line and one from circle.

Total required cases \(= 8C4- (4C4+4C3*4C1) = 53\)

Answer = B

Harshgmat wrote:
Of the 8 points in a given plane 4 are on circle O and remaining 4 are on line L. Number of different quadrilaterals that can be formed by these 8 points are

A) 52

B) 53

C) 54

D) 64

E) 74

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Of the 8 points in a given plane 4 are on circle O -Geometry - PS  [#permalink]

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New post 25 Aug 2018, 09:57
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pushpitkc wrote:

Therefore, the total number of quadrilaterals that can be formed by these points are 1+16+36 = 52(Option B)

pushpitkc , your post has a typo. Your total says 52. Total should say 53 to match your answer of Option B.
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Of the 8 points in a given plane 4 are on circle O -Geometry - PS  [#permalink]

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New post 25 Aug 2018, 10:07
generis wrote:
pushpitkc wrote:

Therefore, the total number of quadrilaterals that can be formed by these points are 1+16+36 = 52(Option B)

pushpitkc , your post has a typo. Your total says 52. Total should say 53 to match your answer of Option B.


Thanks for notifying, made the necessary change!
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Of the 8 points in a given plane 4 are on circle O -Geometry - PS   [#permalink] 25 Aug 2018, 10:07
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