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Math Expert V
Joined: 02 Sep 2009
Posts: 61510
Of the 8 points in a given plane 4 are on circle O and remaining 4 are  [#permalink]

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Difficulty:   65% (hard)

Question Stats: 52% (02:23) correct 48% (01:56) wrong based on 58 sessions

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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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VP  V
Joined: 19 Oct 2018
Posts: 1309
Location: India
Re: Of the 8 points in a given plane 4 are on circle O and remaining 4 are  [#permalink]

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2
= 4C4+4C3*4C1+4C2*4C2
= 1+16+36
=53

Bunuel 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

Are You Up For the Challenge: 700 Level Questions
GMAT Club Legend  V
Joined: 18 Aug 2017
Posts: 5920
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)
Of the 8 points in a given plane 4 are on circle O and remaining 4 are  [#permalink]

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together points of circle + line to form quadrilateral
4c3*4c1 + 4c2 * 4c2
=> 16+ 36
=> 52
IMO A
4c3*4c1 + 4c2 * 4c2+4c4
using all the points possible quadrilateral = 16+ 36+1 ; 53
IMO B

Bunuel 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

Are You Up For the Challenge: 700 Level Questions
VP  P
Joined: 24 Nov 2016
Posts: 1242
Location: United States
Re: Of the 8 points in a given plane 4 are on circle O and remaining 4 are  [#permalink]

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2
Bunuel 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

cases circle + line:
4C2*4C2=36
4C3*4C1=16

cases circle only:
4C4=1

total: 36+16+1=53

Ans (B)
Target Test Prep Representative V
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 9487
Location: United States (CA)
Re: Of the 8 points in a given plane 4 are on circle O and remaining 4 are  [#permalink]

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2
2
Bunuel 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

Are You Up For the Challenge: 700 Level Questions

The four points on circle O are not collinear; however, the four points on line L are. The three methods to form a quadrilateral, using four of the eight points, are:

1) All four points are from circle O,

2) Three points are from circle O, and one point is from line L, and

3) Two points are from circle O, and two points are from line L.

Let’s now determine the number of ways for each method.

1) All four points are from circle O

The number of ways using this method is 4C4 = 1.

2) Three points from circle O and one point from line L

The number of ways using this method is 4C3 x 4C1 = 4 x 4 = 16.

3) Two points from circle O and two points from line L

The number of ways using this method is 4C2 x 4C2 = 6 x 6 = 36.

Therefore, the total number of quadrilaterals that can be formed is 1 + 16 + 36 = 53.

Alternate Solution:

From a total of 8 points, 4 points can be chosen in 8C4 = 8!/(4!*4!) = 70 ways.

Among these 70 choices, the following will not form a quadrilateral:

1) All the four points are from line L. There’s only one such choice.

2) Three points are from line L and one point from circle O: The three points can be chosen from the line in 4C3 = 4 ways. The one point can be chosen from the circle in 4C1 = 4 ways. Thus, there are 4 x 4 = 16 such choices.

We see that 16 + 1 = 17 of the 70 choices of four points do not form a quadrilateral and 70 - 17 = 53 choices do.

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# Scott Woodbury-Stewart

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Scott@TargetTestPrep.com

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If you find one of my posts helpful, please take a moment to click on the "Kudos" button. Re: Of the 8 points in a given plane 4 are on circle O and remaining 4 are   [#permalink] 02 Dec 2019, 18:12
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# Of the 8 points in a given plane 4 are on circle O and remaining 4 are  