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Let T_n be the number of all possible triangles formed by

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

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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
[Reveal] Spoiler: OA

Last edited by Bunuel on 26 Mar 2014, 07:19, edited 1 time in total.
Renamed the topic and edited the question.

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

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New post 26 Mar 2014, 07:21
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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 \(T_{n+1} - T_n =C^3_{n+1}-C^3_n= 10\) --> \(C^3_{n+1}-C^3_n= \frac{(n+1)!}{3!(n-2)!}-\frac{n!}{3!(n-3)!}=10\) --> \(\frac{(n-1)n(n+1)}{6}-\frac{(n-2)(n-1)n}{6}=10\) --> \((n-1)n=20\) --> \(n=5\).

Answer: A.

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 --> \(C^2_n=10\) --> \((n-1)n=20\) --> \(n=5\).

Answer: A.

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

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