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# X persons stand on the circumference of a circle at distinct points.

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Intern
Joined: 29 Jun 2018
Posts: 15
Location: India
GMAT 1: 530 Q42 V26
GPA: 4
X persons stand on the circumference of a circle at distinct points.  [#permalink]

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Updated on: 09 Aug 2018, 23:34
4
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Difficulty:

25% (medium)

Question Stats:

81% (01:57) correct 19% (01:53) wrong based on 79 sessions

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X persons stand on the circumference of a circle at distinct points. Each possible pair of persons, not standing next to each other, sings a two-minute song one pair after the other. If the total time taken for singing is 28 minutes, what is X?

(a) 5
(b) 7
(c) 9
(d) 4
(e) 11

Originally posted by LMP on 09 Aug 2018, 07:20.
Last edited by Bunuel on 09 Aug 2018, 23:34, edited 1 time in total.
Renamed the topic.
Math Expert
Joined: 02 Aug 2009
Posts: 7098
Re: X persons stand on the circumference of a circle at distinct points.  [#permalink]

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17 Aug 2018, 10:24
2
Bulusuchaitanya wrote:
X persons stand on the circumference of a circle at distinct points. Each possible pair of persons, not standing next to each other, sings a two-minute song one pair after the other. If the total time taken for singing is 28 minutes, what is X?

(a) 5
(b) 7
(c) 9
(d) 4
(e) 11

A way to look at it..
When you join these X person, a polygon is formed. And as we do not count the neighbors, it is basically asking us to find DIAGONAL..
Number of diagonals = X(x-3)/2=28/2....X(X-3)=28=7*4..
Thus X is 7

another way...
A person cannot make pair with himself and his neighbours so 3 out of X gone..
So he can make pair with X-3 person..
Thus total X*X-3 but this includes where each combination is counted twice A with C as a different way from C with A..
So total =X(X-3)/2
Total minutes=28, so pairs = 28/2=14..
Thus X (X-3)/2=14....X(X-3)=28=7*4
X=7
B
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##### General Discussion
Intern
Joined: 03 Apr 2017
Posts: 45
Re: X persons stand on the circumference of a circle at distinct points.  [#permalink]

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09 Aug 2018, 08:25
What is the number of pairs here?

According to me it is X(X-3): [Rationale: X-3 as we substract possible number of pairants by selecting any one, and not consider the other 2 people next to him on either side; X as that is the total number of starting persons we can repeat this exercise with]

In this case my equation comes: X(X-3)*2 = 28

And X is not any of the answers. Can someone help me by pointing out where I went wrong.
Intern
Joined: 29 Jun 2018
Posts: 15
Location: India
GMAT 1: 530 Q42 V26
GPA: 4
Re: X persons stand on the circumference of a circle at distinct points.  [#permalink]

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09 Aug 2018, 08:36
1
mitrakaushi wrote:
What is the number of pairs here?

According to me it is X(X-3): [Rationale: X-3 as we substract possible number of pairants by selecting any one, and not consider the other 2 people next to him on either side; X as that is the total number of starting persons we can repeat this exercise with]

In this case my equation comes: X(X-3)*2 = 28

And X is not any of the answers. Can someone help me by pointing out where I went wrong.

So Each pair is getting calculated twice in your Equation. You may have to divide the Equation with 2.
then the equation becomes x(x-3)= 28
and the answer you get is 7

Now what I really dont know is if this is a Hard question or a medium level question.
Intern
Joined: 15 May 2018
Posts: 4
X persons stand on the circumference of a circle at distinct points.  [#permalink]

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17 Aug 2018, 08:27
Hi guys anyone can explain this better i'm struggling to answer it,nor am i able to understand the formula's posted above.
Any experts able to throw some light.
Manager
Joined: 18 Jun 2018
Posts: 249
X persons stand on the circumference of a circle at distinct points.  [#permalink]

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17 Aug 2018, 09:35
4
Avinasht123 wrote:
Hi guys anyone can explain this better i'm struggling to answer it,nor am i able to understand the formula's posted above.
Any experts able to throw some light.

Avinasht123
I am not expert, I can try to explain

Let there be $$n$$ person standing in a circle. Then we have to form teams with 2 people each, such that these 2 people selected in a team should not be adjacent to each other in the circle.

For first person , we can $$n$$ choices, we can select anybody out of $$n$$ persons.

For second person, we have $$n-3$$ [$$n- 1$$(person already selected)$$-1$$(Person to the right of already selected person)$$-1$$(Person to the left of already selected person)]

Total number of such 2 person team:$$\frac{n(n-3)}{2}$$

Division by $$2$$ is required as expression $$n(n-3)$$ counts AB,BA as two pairs , whereas they are only one pair

Now Question says each team formed should sing a 2 minute song.

Total time spent by $$\frac{n(n-3)}{2}$$ teams singing=$$2* \frac{n(n-3)}{2}$$minutes

Now in question , it is given that $$n(n-3)=28$$
This leads to $$n(n-3)=28$$
$$n^2-3n-28=0$$
$$n^2-7n+4n-28=0$$
$$n(n-7)+4(n-7)=0$$
$$(n+4)(n-7)=0$$
$$n=7$$ neglecting $$n=-4$$
Intern
Joined: 15 May 2018
Posts: 4
X persons stand on the circumference of a circle at distinct points.  [#permalink]

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20 Aug 2018, 06:26
Bismark thank you very much for the explanation are you sure you are not an expert
Chetan2u thanks a ton.
Intern
Joined: 03 Nov 2018
Posts: 7
Re: X persons stand on the circumference of a circle at distinct points.  [#permalink]

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07 Dec 2018, 11:53
Is there a way to solve this with circular combinations?
Re: X persons stand on the circumference of a circle at distinct points. &nbs [#permalink] 07 Dec 2018, 11:53
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