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# Which of the following could be the number of diagonals of an n-polygo

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Which of the following could be the number of diagonals of an n-polygo  [#permalink]

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12 Mar 2019, 00:43
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65% (hard)

Question Stats:

49% (01:46) correct 51% (01:12) wrong based on 35 sessions

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[GMAT math practice question]

Which of the following could be the number of diagonals of an n-polygon?

A. 3
B. 6
C. 8
D. 12
E. 20

_________________
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"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Manager Joined: 19 Jan 2019 Posts: 110 Re: Which of the following could be the number of diagonals of an n-polygo [#permalink] ### Show Tags 12 Mar 2019, 10:31 sameeruce08 wrote: Wrong question Posted from my mobile device I think the question is asking the maximum number of diagonals a polygon can have.... An octagon has 20 diagonals, that's why the OA is E.... the next polygon is enneagon with 27 diagonals.... But the maximum number given in answer choices is 20... And that's why it's E .. Intern Joined: 11 Apr 2017 Posts: 21 Location: India Concentration: General Management, Strategy Schools: ISB '21, IIMA , IIMB, IIMC , XLRI GPA: 4 WE: General Management (Energy and Utilities) Re: Which of the following could be the number of diagonals of an n-polygo [#permalink] ### Show Tags 12 Mar 2019, 11:03 An 8 sided polygon will havde 20 diagonal diagonal of a polygon=n(n-3)/2 put integers values of n from 4 to 8 only option E is possible IMO-E GMATH Teacher Status: GMATH founder Joined: 12 Oct 2010 Posts: 935 Which of the following could be the number of diagonals of an n-polygo [#permalink] ### Show Tags 12 Mar 2019, 15:53 MathRevolution wrote: [GMAT math practice question] Which of the following could be the number of diagonals of an n-polygon? A. 3 B. 6 C. 8 D. 12 E. 20 $$?\,\,\,:\,\,\,\# \,d\,\,\underline {{\rm{could}}\,\,{\rm{be}}} \,\,\,\,\left( {N \ge 3\,\,{\mathop{\rm int}} \,\,\left( * \right)\,,\,\,N{\rm{ - polygon}}} \right)$$ $$d\,\, \to \,\,\,\left\{ \matrix{ \,{\rm{each}}\,\,{\rm{vertex}}\,\,{\rm{with}}\,\,\left( {{\rm{not}}\,\,{\rm{itself}}} \right)\,\,{\rm{nor}}\,\,\left( {{\rm{next \,\, to}}\,\,{\rm{it}}\,\,{\rm{vertex}}} \right) \hfill \cr \,\left[ {A - C} \right]\,\,{\rm{diagonal}}\,\,{\rm{is}}\,\,{\rm{the}}\,\,{\rm{same}}\,\,{\rm{of}}\,\,\left[ {C - A} \right]\,\,{\rm{diagonal}} \hfill \cr} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,d = {{N\left( {N - 3} \right)} \over 2}$$ Alternate argument: (Explain!) $$d = C\left( {N,2} \right) - N = {{N!} \over {2!\left( {N - 2} \right)!}} - N = {{N\left( {N - 1} \right)} \over 2} - {{2N} \over 2} = {{N\left( {N - 3} \right)} \over 2}$$ $$\left. \matrix{ \left( A \right)\,\,\,{{N\left( {N - 3} \right)} \over 2} = 3\,\,\,\,\, \Rightarrow \,\,\,\,\,N\left( {N - 3} \right) = 6\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,{\rm{impossible}} \hfill \cr \left( B \right)\,\,\,{{N\left( {N - 3} \right)} \over 2} = 6\,\,\,\,\, \Rightarrow \,\,\,\,\,N\left( {N - 3} \right) = 12\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,{\rm{impossible}} \hfill \cr \left( C \right)\,\,\,{{N\left( {N - 3} \right)} \over 2} = 8\,\,\,\,\, \Rightarrow \,\,\,\,\,N\left( {N - 3} \right) = 16\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,{\rm{impossible}} \hfill \cr \left( D \right)\,\,\,{{N\left( {N - 3} \right)} \over 2} = 12\,\,\,\,\, \Rightarrow \,\,\,\,\,N\left( {N - 3} \right) = 24\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,{\rm{impossible}}\,\,\, \hfill \cr} \right\}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( E \right)\,\,{\rm{by}}\,\,{\rm{exclusion}}\,\,\,\,\left( {**} \right)$$ $$\left( {**} \right)\,\,\,\,\left( E \right)\,\,\,{{N\left( {N - 3} \right)} \over 2} = 20\,\,\,\,\, \Rightarrow \,\,\,\,\,N\left( {N - 3} \right) = 40\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,N = 8$$ We follow the notations and rationale taught in the GMATH method. Regards, Fabio. _________________ Fabio Skilnik :: GMATH method creator (Math for the GMAT) Our high-level "quant" preparation starts here: https://gmath.net Senior Manager Joined: 27 Dec 2016 Posts: 309 Re: Which of the following could be the number of diagonals of an n-polygo [#permalink] ### Show Tags 12 Mar 2019, 18:33 Number of diagonals can be found using n(n-3)/2 formula. Except E, all options fail to satisfy this equation. Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 8235 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: Which of the following could be the number of diagonals of an n-polygo [#permalink] ### Show Tags 14 Mar 2019, 01:52 => The number of diagonals of an n-polygon is nC2 – n = $$\frac{n(n-1)}{2} – n=\frac{n(n-3)}{2}.$$ If $$n = 4$$, then the number of diagonals is 4C2 – 4 = 6 – 4 = 2. If $$n = 5$$, then the number of diagonals is 5C2 – 5 = 10 – 5 = 5. If $$n = 6$$, then the number of diagonals is 6C2 – 6 = 15 – 6 = 9. If $$n = 7$$, then the number of diagonals is 7C2 – 7 = 21 – 7 = 14. If $$n = 8$$, then the number of diagonals is 8C2 – 8 = 28 – 8 = 20. Therefore, E is the answer. Answer: E _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 1 month Online Course"
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Re: Which of the following could be the number of diagonals of an n-polygo  [#permalink]

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17 Mar 2019, 23:06
MathRevolution wrote:
[GMAT math practice question]

Which of the following could be the number of diagonals of an n-polygon?

A. 3
B. 6
C. 8
D. 12
E. 20

diagonal of polygon ; n * (n-3)/2
so solve using given options
say
20= n*(n-3)/2
we get = n= 8 ,-5
n = 8 is max possible value of n
IMO E
Re: Which of the following could be the number of diagonals of an n-polygo   [#permalink] 17 Mar 2019, 23:06
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