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# Let T_n be the number of all possible triangles formed by

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Manager
Joined: 04 Oct 2013
Posts: 155
Location: India
GMAT Date: 05-23-2015
GPA: 3.45
Let T_n be the number of all possible triangles formed by  [#permalink]

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Updated on: 26 Mar 2014, 07:19
2
5
00:00

Difficulty:

85% (hard)

Question Stats:

55% (01:37) correct 45% (02:04) wrong based on 132 sessions

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Let $$T_n$$ be the number of all possible triangles formed by joining vertices of an n-sided regular polygon. If $$T_{n+1} - T_n = 10$$, then the value of n is

A. 5
B. 6
C. 7
D. 8
E. 10

Originally posted by arunspanda on 26 Mar 2014, 06:17.
Last edited by Bunuel on 26 Mar 2014, 07:19, edited 1 time in total.
Renamed the topic and edited the question.
Math Expert
Joined: 02 Sep 2009
Posts: 47977
Re: Let T_n be the number of all possible triangles formed by  [#permalink]

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26 Mar 2014, 07:21
arunspanda wrote:
Let $$T_n$$ be the number of all possible triangles formed by joining vertices of an n-sided regular polygon. If $$T_{n+1} - T_n = 10$$, then the value of n is

A. 5
B. 6
C. 7
D. 8
E. 10

In a plane if there are n points of which no three are collinear, then the number of triangles that can be formed by joining them is $$C^3_n$$.

We are given that $$T_{n+1} - T_n =C^3_{n+1}-C^3_n= 10$$ --> $$C^3_{n+1}-C^3_n= \frac{(n+1)!}{3!(n-2)!}-\frac{n!}{3!(n-3)!}=10$$ --> $$\frac{(n-1)n(n+1)}{6}-\frac{(n-2)(n-1)n}{6}=10$$ --> $$(n-1)n=20$$ --> $$n=5$$.

Or: one additional point gives 10 more triangles, so when we add one point there are 10 different pairs of points which make triangles with that additional point --> $$C^2_n=10$$ --> $$(n-1)n=20$$ --> $$n=5$$.

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Re: Let T_n be the number of all possible triangles formed by  [#permalink]

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25 Oct 2017, 23:59
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Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: Let T_n be the number of all possible triangles formed by &nbs [#permalink] 25 Oct 2017, 23:59
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