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

Question

N 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 N?

A. 4
B. 5
C. 6
D. 7
E. 8

A. 4
B. 5 
C. 6
D. 7
E. 8

chetan2u · Answer

Join the position of all of the persons, and  you will get a polygon of N sides.

Now  we are basically looking at choosing a diagonal except the sides of polygon  => nC2-n
nC2 is the number of ways we can choose a pair from n person.... It is same as the number of ways lines can be drawn in a polygon of n sides. 
n is the number of ways of pairs that consists of person standing next to each other.... It is same as the number of sides of the polygon.

This should be equal to 28/2=14, as each pair sang a song for 2 minutes. => nC2-n=14
 \frac{n(n-1)}{2}-n=14.........n^2-n-2n=28......n(n-3)=28=7*4 
So, n=7.

D

chetan2u · Answer

nC2 is the number of ways we can choose a pair from n person.... It is same as the number of ways lines can be drawn in a polygon of n sides. 
n is the number of ways of pairs that consists of person standing next to each other.... It is same as the number of sides of the polygon.

abhinavsodha800 · Answer

Hi Chetan, i have few doubts
1. how did you come to know that it will be a polygon (In the question it is give as standing on circle so why polygon )?

chetan2u · Answer

Hi

If you draw a line between each adjoining pair, you will have a polygon equal to the number of people. This is just to visualize the pairs.
I have attached a figure for reference.

abhinavsodha800 · Answer

ok i got it . one more question is that can it be polygon of more than 5 sides because it is mentioned n people ?

chetan2u · Answer

It could be anything - 7, 700 anything 
We get the value of n from nC2-n=28/2

PgRaul · Answer

ANs : E
It is taking about pair of people. So 8 people, 4 pairs each, singing 2 mins each, 7 times. 
#expertsreply

pudu · Answer

Is it possible to get more questions of this type?

ruis · Answer

Is there another approach to this question without polygons? I get the following:

Total time: 28 minutes
28/2 = 14 pairs singing
From that I would have done nC2 = 14... Which leads to nowhere. I still struggle to understand nC2 - n = 14. Geometry is not my strength.
How can I anchor the restriction here? "Not standing next to each other" You always have 2 people next....

Favorable outcome / all possible ourcomes...14 is my favourable outcome right? I am getting super confused here.

Also, as they stand on a circunference, should we take into account circular arrangement type of Permutation? (n-1)!? Instead con Combination?

Is this a quality question? Should I stick only to practicing only official questions?

ruis · Answer

It took me a long time to understand that te restriction is -n. I got there by drawing difference scenarios. Still struggling with solvimg fast the green writing… I would just plug numbers and run

However I would have never been able to answer this qq under 2 mins. Would love to read other reasonings! Thank you everyone!!!

Posted from my mobile device

Regor60 · Answer

First recognize that the number of pairs without restriction is

N!/2!(N-2)! = N(N-1)/2

The restriction is no pairs of people next to each other.

So just imagine them in a line of, say, 4 people.

The first person is paired with the second, second with third and third with fourth.

How many pairs is that ?

4-1 = 3, which is N-1.

But, in a circle, the last person can also be paired with the first, so that's another pair, so the total disallowed pairs is:

N

Answering the question then becomes:

{ -N}*2 = 28, or:

N^2-3N-28 = 0.

This factors into:

(N-7)(N+4) = 0 and:

N=7

Visualizing polygons is completely unnecessary.

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

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards