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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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New post 12 Mar 2019, 00:43
00:00
A
B
C
D
E

Difficulty:

  75% (hard)

Question Stats:

44% (01:49) correct 56% (01:12) wrong based on 32 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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Re: Which of the following could be the number of diagonals of an n-polygo  [#permalink]

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New post 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 ..
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Re: Which of the following could be the number of diagonals of an n-polygo  [#permalink]

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

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New post 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.
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Re: Which of the following could be the number of diagonals of an n-polygo  [#permalink]

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New post 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.
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Re: Which of the following could be the number of diagonals of an n-polygo  [#permalink]

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

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

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