It is currently 19 Apr 2018, 19:09

### 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

# How many unique quadrilaterals can be inscribed in the verti

Author Message
TAGS:
Senior Manager
Joined: 20 May 2014
Posts: 282
Followers: 9

Kudos [?]: 41 [0], given: 219

How many unique quadrilaterals can be inscribed in the verti [#permalink]  12 Nov 2017, 01:18
00:00

Question Stats:

0% (00:00) correct 100% (01:54) wrong based on 1 sessions
How many unique quadrilaterals can be inscribed in the vertices of a nonagon (a 9-sided figure), if points A and B, two vertices in the nonagon, cannot make up the same quadrilateral?

(A) 126

(B) 105

(C) 96

(D) 65

(E) 21

Kudos for correct solution.
[Reveal] Spoiler: OA
Director
Joined: 03 Sep 2017
Posts: 521
Followers: 0

Kudos [?]: 273 [1] , given: 66

Re: How many unique quadrilaterals can be inscribed in the verti [#permalink]  13 Nov 2017, 00:24
1
KUDOS
The total number of quadrilaterals that can be inscribed in a nonagon can be computed by counting the number of ways 4 points can be chosen out of 9, i.e. 9C4 = \frac{9!}{5!4!} = 126.

But this is not what we are looking for. Indeed, we need to exclude those quadrilaterals which have two points being A and B. To compute them, given that two points are fixed, we can count how many ways are there to choose 2 points out of 7, i.e. 7C2 = \frac{7!}{2!5!} = 21. Subtracting 21 from 126, we get 105.

Target Test Prep Representative
Affiliations: Target Test Prep
Joined: 09 May 2016
Posts: 44
Location: United States
Followers: 3

Kudos [?]: 34 [1] , given: 0

Re: How many unique quadrilaterals can be inscribed in the verti [#permalink]  19 Dec 2017, 06:32
1
KUDOS
Expert's post
Bunuel wrote:
How many unique quadrilaterals can be inscribed in the vertices of a nonagon (a 9-sided figure), if points A and B, two vertices in the nonagon, cannot make up the same quadrilateral?

(A) 126

(B) 105

(C) 96

(D) 65

(E) 21

Since a nonagon has 9 vertices and a quadrilateral has 4 vertices, the number of quadrilaterals that can be made is 9C4 = (9 x 8 x 7 x 6)/4! = (9 x 8 x 7 x 6)/(4 x 3 x 2) = 3 x 7 x 6 = 126, if there are no restrictions. However, since vertices A and B can’t both be in the same quadrilateral, we need to subtract the number of quadrilaterals that have both vertices A and B. The number of such quadrilaterals is 2C2 x 7C2 = 1 x (7 x 6)/2! = 42/2 = 21 (notice that 2C2 is the number of ways A and B can be picked if they have to be 2 vertices of the quadrilateral and 7C2 is the number of ways the other 2 vertices of the quadrilateral can be picked from the other 7 vertices).

Thus, the number of quadrilaterals such that vertices A and B are not in the same quadrilateral is 126 - 21 = 105.

_________________

Jeffrey Miller
Jeffrey Miller

Re: How many unique quadrilaterals can be inscribed in the verti   [#permalink] 19 Dec 2017, 06:32
Display posts from previous: Sort by