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Thirteen distinct points lie in a plane with exactly n of the points

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Joined: 02 Sep 2009
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Thirteen distinct points lie in a plane with exactly n of the points  [#permalink]

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02 Dec 2019, 00:39
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Difficulty:

95% (hard)

Question Stats:

33% (02:35) correct 67% (02:36) wrong based on 43 sessions

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Thirteen distinct points lie in a plane with exactly n of the points lying on the same line and the other 13-n points non-collinear. If 276 unique triangles can be drawn using these points as vertices, what is the value of n?

A. 2
B. 3
C. 4
D. 5
E. 7

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Re: Thirteen distinct points lie in a plane with exactly n of the points  [#permalink]

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02 Dec 2019, 02:43
1
total ∆ ; 13c3 ; 286
and since 276 unique ∆ are to be formed ; 286-nc3 ; 276
n has to be 5
IMO D

Bunuel wrote:
Thirteen distinct points lie in a plane with exactly n of the points lying on the same line and the other 13-n points non-collinear. If 276 unique triangles can be drawn using these points as vertices, what is the value of n?

A. 2
B. 3
C. 4
D. 5
E. 7

Are You Up For the Challenge: 700 Level Questions
Director
Joined: 27 May 2012
Posts: 945
Thirteen distinct points lie in a plane with exactly n of the points  [#permalink]

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18 Dec 2019, 09:55
Bunuel wrote:
Thirteen distinct points lie in a plane with exactly n of the points lying on the same line and the other 13-n points non-collinear. If 276 unique triangles can be drawn using these points as vertices, what is the value of n?

A. 2
B. 3
C. 4
D. 5
E. 7

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$$n_{c_1} *{(13-n)}_{c_2} +n_{c_2} *{(13-n)}_{c_1}+{(13-n)}_{c_3}$$=276

Where $$n_{c_1}$$ = Choose 1 point from the n collinear points
$${(13-n)}_{c_2}$$= Choose 2 points from the (13-n ) NON-collinear points
$${(13-n)}_{c_3}$$= Choose 3 points from the (13-n) NON-collinear points

It is better to substitute answer choices after this , we see that for n = 5 , the above equation gives 276

Ans-D

Hope it's clear.
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Re: Thirteen distinct points lie in a plane with exactly n of the points  [#permalink]

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23 Dec 2019, 18:34
1
Bunuel wrote:
Thirteen distinct points lie in a plane with exactly n of the points lying on the same line and the other 13-n points non-collinear. If 276 unique triangles can be drawn using these points as vertices, what is the value of n?

A. 2
B. 3
C. 4
D. 5
E. 7

Are You Up For the Challenge: 700 Level Questions

If all 13 points are non-collinear, then 13C3 = (13 x 12 x 11) / (3 x 2) = 13 x 2 x 11 = 286 unique triangles can be drawn. However, since only 276 triangles can be drawn, 10 triangles must have been created by the n collinear points. Therefore, we can create the equation:

nC3 = 10

n! / [3! x (n - 3)!]

n(n - 1)(n - 2) / (3 x 2) = 10

n(n - 1)(n - 2) = 60

From the above equation, we see n must be 5 since 5(4)(3) = 60.

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Re: Thirteen distinct points lie in a plane with exactly n of the points   [#permalink] 23 Dec 2019, 18:34
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