How many diagonals does a polygon with 21 sides have, if one : GMAT Problem Solving (PS)
Check GMAT Club Decision Tracker for the Latest School Decision Releases http://gmatclub.com/AppTrack

 It is currently 20 Jan 2017, 15:44

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# How many diagonals does a polygon with 21 sides have, if one

Author Message
TAGS:

### Hide Tags

Intern
Joined: 30 Aug 2010
Posts: 13
Followers: 0

Kudos [?]: 32 [2] , given: 2

How many diagonals does a polygon with 21 sides have, if one [#permalink]

### Show Tags

23 Sep 2010, 01:59
2
KUDOS
11
This post was
BOOKMARKED
00:00

Difficulty:

65% (hard)

Question Stats:

64% (02:12) correct 36% (01:25) wrong based on 199 sessions

### HideShow timer Statistics

How many diagonals does a polygon with 21 sides have, if one of its vertices does not connect to any diagonal?

A. 21
B. 170
C. 340
D. 357
E. 420

[Reveal] Spoiler:
I was considering a different approach of 20 vertices (21 vertices -1 vertex not linking) linking to 17 (21 vertices - 1 vertix I'm considering -2 vertices next to the one I'm considering - 1 vertex not linking) others each: that is 17*20=340. We divide that by 2 since every diagonal connects two vertices. 340/2=170.

Consider the parallel case in which we have 21 people who shake hands to each other.
In that case we'll have $$21*20/2$$ handshakes.
This because, if we consider all the 21 people in a line, the first one will shake hands to 20, the second to 19, the third to 18 people and so on. So we have:
20+19+18+...+1
This sum equals to $$21*20/2=210$$.

Handshaking and diagonals are different in counting since diagonals don't include sides.
To calculate diagonals we have to subtract the number of sides (that are 21) from $$21*20/2$$.
So:
N° of diagonals = $$(21*20/2)-21=210-21=189$$

Since one vertex does not connect to any diagonals, we have to subtract the diagonals of this vertex.
Diagonals of one vertex equals to: 21 (number of vertex) - 1 (itself) - 2 (sides with vertices next to it).

So the answer should be: $$189-18=171$$
That's different from 170 obtained above.

What's wrong with my second approach?
[Reveal] Spoiler: OA
Math Expert
Joined: 02 Sep 2009
Posts: 36583
Followers: 7087

Kudos [?]: 93280 [1] , given: 10555

Re: How many diagonals does a polygon with 21 sides have... [#permalink]

### Show Tags

23 Sep 2010, 02:49
1
KUDOS
Expert's post
3
This post was
BOOKMARKED
rraggio wrote:
How many diagonals does a polygon with 21 sides have, if one of its vertices does not connect to any diagonal?

a) 21
b) 170
c) 340
d) 357
e) 420

I don't think B (170) is a correct answer.

Generally the # of diagonals in $$n$$ sided polygon equals to $$C^2_n-n=\frac{n(n-3)}{2}$$: $$C^2_n$$ choose any two vertices out of $$n$$ to connect minus $$n$$ sides, which won't be diagonals.

So, # of diagonals in 21 sided polygon is $$C^2_n-n=210-21=189$$. Since the diagonals from 1 particulat vertex shouldn't be counted then $$189-(21-3)=171$$ (one vertex makes $$n-3$$ diagonals).

_________________
Retired Moderator
Joined: 02 Sep 2010
Posts: 805
Location: London
Followers: 105

Kudos [?]: 957 [2] , given: 25

Re: How many diagonals does a polygon with 21 sides have... [#permalink]

### Show Tags

23 Sep 2010, 03:04
2
KUDOS
2
This post was
BOOKMARKED
Take any vertex in the polygon. To form a diagnol, it can be connected to any one of 18 vertices (counting out itself, and the 2 vertices adjacent to it).

If we count diagnols like this, we will double count everything (once for every end point). So the total number of diagnols = (18*21)/2 = 189

Each vertex has 18 diagnols, so if we take all the diagnols from this one vertex out .. we are left with 171 diagnols

Alternate Solution

Consider a polygon with 20 sides. By logic similar to above, the number of diagnols is (17*20)/2 = 170

Now take a new point X and any side of the 20-sided figure AB. Join XA and XB to form a 21 sided figure. In this new figure all the old diagnols are still diagnols + the side AB also becomes a diagnol. Also note that there are no diagnols originating from X

Hence the number of diagnols = 170 + 1 (AB) = 171 diagnols
_________________
Math Expert
Joined: 02 Sep 2009
Posts: 36583
Followers: 7087

Kudos [?]: 93280 [0], given: 10555

Re: How many diagonals does a polygon with 21 sides have... [#permalink]

### Show Tags

23 Sep 2010, 03:05
rraggio wrote:
I was considering a different approach of 20 vertices (21 vertices -1 vertex not linking) linking to 17 (21 vertices - 1 vertix I'm considering -2 vertices next to the one I'm considering - 1 vertex not linking) others each: that is 17*20=340. We divide that by 2 since every diagonal connects two vertices. 340/2=170.

Consider the parallel case in which we have 21 people who shake hands to each other.
In that case we'll have $$21*20/2$$ handshakes.
This because, if we consider all the 21 people in a line, the first one will shake hands to 20, the second to 19, the third to 18 people and so on. So we have:
20+19+18+...+1
This sum equals to $$21*20/2=210$$.

Handshaking and diagonals are different in counting since diagonals don't include sides.
To calculate diagonals we have to subtract the number of sides (that are 21) from $$21*20/2$$.
So:
N° of diagonals = $$(21*20/2)-21=210-21=189$$

Since one vertex does not connect to any diagonals, we have to subtract the diagonals of this vertex.
Diagonals of one vertex equals to: 21 (number of vertex) - 1 (itself) - 2 (sides with vertices next to it).

So the answer should be: $$189-18=171$$
That's different from 170 obtained above.

What's wrong with my second approach?

Actually first approach is wrong: 17*20 is not right. As 2 vertices which are next to excluded vertex can be connected each to 21-3=18 vertices, so 2*18. Other 18 vertices can be connected to 21 - 3 - 1 (excluded vertex) = 17 vertices. So we would have 2*18+18*17=342 --> divided by 2 to exclude double counting --> 342/2=171.
_________________
CEO
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2795
Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35
Followers: 226

Kudos [?]: 1620 [1] , given: 235

Re: How many diagonals does a polygon with 21 sides have... [#permalink]

### Show Tags

23 Sep 2010, 05:47
1
KUDOS
shrouded1 wrote:

Consider a polygon with 20 sides. By logic similar to above, the number of diagnols is (17*20)/2 = 170

Now take a new point X and any side of the 20-sided figure AB. Join XA and XB to form a 21 sided figure. In this new figure all the old diagnols are still diagnols + the side AB also becomes a diagnol. Also note that there are no diagnols originating from X

Hence the number of diagnols = 170 + 1 (AB) = 171 diagnols

I did same way and got 170 forgot to add +1. Thanks !!
_________________

Fight for your dreams :For all those who fear from Verbal- lets give it a fight

Money Saved is the Money Earned

Jo Bole So Nihaal , Sat Shri Akaal

GMAT Club Premium Membership - big benefits and savings

Gmat test review :
http://gmatclub.com/forum/670-to-710-a-long-journey-without-destination-still-happy-141642.html

Intern
Joined: 11 Sep 2010
Posts: 3
Followers: 0

Kudos [?]: 0 [0], given: 0

Polygon problems a sucking combination---HELP! [#permalink]

### Show Tags

10 Nov 2010, 05:59
Hey Guys, I need some help out here...I picked up this question from a list of questions posted by a fellow gmat geek.
I am not able to figure out out how to solve this question in terms of whether I should I directly work out a 20c2 for this question.

To sum it up please let me know how to solve problems like these in terms of what should be the basic approach.

How many diagonals does a polygon with 21 sides have, if one of its vertices does not connect to any diagonal?

21
170
340
357
420
Intern
Joined: 11 Sep 2010
Posts: 3
Followers: 0

Kudos [?]: 0 [0], given: 0

Re: How many diagonals does a polygon with 21 sides have... [#permalink]

### Show Tags

10 Nov 2010, 06:32
Thanks guys! I appreciate your quick response!
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7125
Location: Pune, India
Followers: 2137

Kudos [?]: 13673 [5] , given: 222

Re: How many diagonals does a polygon with 21 sides have... [#permalink]

### Show Tags

10 Nov 2010, 07:17
5
KUDOS
Expert's post
5
This post was
BOOKMARKED
It is good to remember that a polygon with n sides has nC2 - n diagonals and follow up from there as Bunuel did or follow shrouded1's approach, which I thought was pretty cool too.

Though, remember that if you are lost with the 21 sides, cannot think of the formula and are hard pressed for time, try and work it on a smaller scale.
Lets see what happens in a 6 sided figure which is extremely easy to draw:
6 sided figure. 1 vertex left alone. When you start from the first point, you cant join it to 3 of the 6 points - itself, point left alone and point next to it.
Total diagonals drawn: 3 + 2 + 1
Attachment:

Ques.jpg [ 18.05 KiB | Viewed 10055 times ]

21 sided figure. When you will start with the first point, you will not join it to 3 of the 21 points - itself, point left alone and point next to it.
Total diagonals drawn will be: 18 + 17 + 16 + ... +1 = 18*19/2 = 171
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews GMAT Club Legend Joined: 09 Sep 2013 Posts: 13468 Followers: 575 Kudos [?]: 163 [0], given: 0 Re: How many diagonals does a polygon with 21 sides have, if one [#permalink] ### Show Tags 12 Oct 2013, 04:22 Hello from the GMAT Club BumpBot! 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). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________ Intern Joined: 04 Sep 2012 Posts: 4 Followers: 0 Kudos [?]: 0 [0], given: 1 Re: How many diagonals does a polygon with 21 sides have... [#permalink] ### Show Tags 10 Dec 2013, 19:46 Bunuel wrote: rraggio wrote: How many diagonals does a polygon with 21 sides have, if one of its vertices does not connect to any diagonal? a) 21 b) 170 c) 340 d) 357 e) 420 I don't think B (170) is a correct answer. Generally the # of diagonals in $$n$$ sided polygon equals to $$C^2_n-n=\frac{n(n-3)}{2}$$: $$C^2_n$$ choose any two vertices out of $$n$$ to connect minus $$n$$ sides, which won't be diagonals. So, # of diagonals in 21 sided polygon is $$C^2_n-n=210-21=189$$. Since the diagonals from 1 particulat vertex shouldn't be counted then $$189-(21-3)=171$$ (one vertex makes $$n-3$$ diagonals). Answer: 171. Why don't we do simple calculation? Let's say there are 21 sides -> there are total 19 "dots" to connect each diagonals So, 19 "dots" and 2 points to connect for each other = 2C19 = 171 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7125 Location: Pune, India Followers: 2137 Kudos [?]: 13673 [0], given: 222 Re: How many diagonals does a polygon with 21 sides have... [#permalink] ### Show Tags 10 Dec 2013, 20:15 Expert's post 2 This post was BOOKMARKED ltkenny wrote: Why don't we do simple calculation? Let's say there are 21 sides -> there are total 19 "dots" to connect each diagonals So, 19 "dots" and 2 points to connect for each other = 2C19 = 171 How do you have 19 dots? If it is a 21 sided figure, it will have 21 vertices. One vertex is not participating in diagonals so you have 20 vertices participating. You select 2 out of these 20 in 20C2 ways. But this includes the 19 sides of the polygon (since one vertex is not participating, 2 sides are already not formed so you only need to deduct 19 sides.) So total no of diagonals will be 20C2 - 19 = 171 _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

Veritas Prep Reviews

Intern
Joined: 04 Sep 2012
Posts: 4
Followers: 0

Kudos [?]: 0 [0], given: 1

Re: How many diagonals does a polygon with 21 sides have... [#permalink]

### Show Tags

10 Dec 2013, 21:11
VeritasPrepKarishma wrote:

How do you have 19 dots? If it is a 21 sided figure, it will have 21 vertices. One vertex is not participating in diagonals so you have 20 vertices participating. You select 2 out of these 20 in 20C2 ways. But this includes the 19 sides of the polygon (since one vertex is not participating, 2 sides are already not formed so you only need to deduct 19 sides.)
So total no of diagonals will be 20C2 - 19 = 171

Wow, it's nice explanation. Your way is much more understandable.
Thanks.
Manager
Joined: 15 Mar 2012
Posts: 71
Followers: 0

Kudos [?]: 5 [0], given: 20

Re: How many diagonals does a polygon with 21 sides have, if one [#permalink]

### Show Tags

26 Jun 2014, 02:22
Bunuel wrote:
rraggio wrote:
How many diagonals does a polygon with 21 sides have, if one of its vertices does not connect to any diagonal?

a) 21
b) 170
c) 340
d) 357
e) 420

I don't think B (170) is a correct answer.

Generally the # of diagonals in $$n$$ sided polygon equals to $$C^2_n-n=\frac{n(n-3)}{2}$$: $$C^2_n$$ choose any two vertices out of $$n$$ to connect minus $$n$$ sides, which won't be diagonals.

So, # of diagonals in 21 sided polygon is $$C^2_n-n=210-21=189$$. Since the diagonals from 1 particulat vertex shouldn't be counted then $$189-(21-3)=171$$ (one vertex makes $$n-3$$ diagonals).

Q: How many diagonals does a polygon with 18 sides have if three of its vertices do not send any diagonal?
A: #lines with (18-3) vertices = 15C2 = 15 x 14 / 2 = 105
3 vertices would form 4 sides => 4 sides not counted in 105 already
Therefore, #diagonals = #lines - #sides = 105 - (18 - 4) = 91 ---------------------------------------> (1)
Is 91 correct?

If I calculate it using the formulae, #diagonals = n (n-3)/2 & Each vertex sends of n-3 diagonals
#desired diagonals = #diagonals with 18 sides - #diagonals 3 vertices would send = 18 (18-3) / 2 - (18-3) * 3 = 135 - 45 = 90 ---------------------------------------> (2)

Could you please tell why I see this difference in the answers (1) and (2)?

Thank you very much in advance.
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7125
Location: Pune, India
Followers: 2137

Kudos [?]: 13673 [0], given: 222

Re: How many diagonals does a polygon with 21 sides have, if one [#permalink]

### Show Tags

29 Jun 2014, 23:21
divineacclivity wrote:

Q: How many diagonals does a polygon with 18 sides have if three of its vertices do not send any diagonal?
A: #lines with (18-3) vertices = 15C2 = 15 x 14 / 2 = 105
3 vertices would form 4 sides => 4 sides not counted in 105 already
Therefore, #diagonals = #lines - #sides = 105 - (18 - 4) = 91 ---------------------------------------> (1)
Is 91 correct?

If I calculate it using the formulae, #diagonals = n (n-3)/2 & Each vertex sends of n-3 diagonals
#desired diagonals = #diagonals with 18 sides - #diagonals 3 vertices would send = 18 (18-3) / 2 - (18-3) * 3 = 135 - 45 = 90 ---------------------------------------> (2)

Could you please tell why I see this difference in the answers (1) and (2)?

Thank you very much in advance.

Yes, 91 is correct.

The second method is not correct because when you subtract each of the 15 diagonals made by the 3 vertices which we need to ignore, you are double counting 1 diagonal. You actually need to subtract only 44 diagonals. Which diagonal are you double counting? The one which connects 2 of the three ignored vertices. Try to make a polygon with a few vertices. Make a few diagonals. Remove 3 vertices next to each other. 2 of the three vertices which have a vertex between them will be joined by a diagonal. When you remove 15 diagonals for each vertex, you are removing that diagonal twice.

Hence, what you need to do is 135 - 44 = 91.

Mind you, we assumed that the vertices which were removed were next to each other. If they are not, the answer would be different since there would be more double counting.
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for \$199

Veritas Prep Reviews

Manager
Joined: 15 Mar 2012
Posts: 71
Followers: 0

Kudos [?]: 5 [0], given: 20

Re: How many diagonals does a polygon with 21 sides have, if one [#permalink]

### Show Tags

04 Jul 2014, 21:52
VeritasPrepKarishma wrote:
divineacclivity wrote:

Q: How many diagonals does a polygon with 18 sides have if three of its vertices do not send any diagonal?
A: #lines with (18-3) vertices = 15C2 = 15 x 14 / 2 = 105
3 vertices would form 4 sides => 4 sides not counted in 105 already
Therefore, #diagonals = #lines - #sides = 105 - (18 - 4) = 91 ---------------------------------------> (1)
Is 91 correct?

If I calculate it using the formulae, #diagonals = n (n-3)/2 & Each vertex sends of n-3 diagonals
#desired diagonals = #diagonals with 18 sides - #diagonals 3 vertices would send = 18 (18-3) / 2 - (18-3) * 3 = 135 - 45 = 90 ---------------------------------------> (2)

Could you please tell why I see this difference in the answers (1) and (2)?

Thank you very much in advance.

Yes, 91 is correct.

The second method is not correct because when you subtract each of the 15 diagonals made by the 3 vertices which we need to ignore, you are double counting 1 diagonal. You actually need to subtract only 44 diagonals. Which diagonal are you double counting? The one which connects 2 of the three ignored vertices. Try to make a polygon with a few vertices. Make a few diagonals. Remove 3 vertices next to each other. 2 of the three vertices which have a vertex between them will be joined by a diagonal. When you remove 15 diagonals for each vertex, you are removing that diagonal twice.

Hence, what you need to do is 135 - 44 = 91.

Mind you, we assumed that the vertices which were removed were next to each other. If they are not, the answer would be different since there would be more double counting.

Wonderful! Thanks a bunch.
Intern
Joined: 26 Aug 2014
Posts: 1
Followers: 0

Kudos [?]: 0 [0], given: 0

Re: How many diagonals does a polygon with 21 sides have, if one [#permalink]

### Show Tags

26 Aug 2014, 11:00
1
This post was
BOOKMARKED
We have 20 vertices(1 vertex is not working) linking to 17 others(1 vertex will make 17 diagonals) each: that is 17*20=340. We divide that by 2 since every diagonal connects two vertices. 340/2=170. The vertex that does not connect to any diagonal is just not counted
Re: How many diagonals does a polygon with 21 sides have, if one   [#permalink] 26 Aug 2014, 11:00
Similar topics Replies Last post
Similar
Topics:
10 How many diagonals does a regular 11 -sided polygon contain? 6 23 Nov 2016, 05:57
3 How many diagonals does a polygon with 18 sides have if three of its 5 16 Oct 2016, 04:53
4 A polygon has 90 diagonals. How many sides does it have? 3 29 Apr 2016, 23:26
10 How many diagonals does a 63-sided convex polygon have? 4 15 Mar 2016, 12:02
2 how many internal diagonals does a decagon (ten sided polygon) have? 3 27 Feb 2015, 02:18
Display posts from previous: Sort by