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

Question

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

Bismarck · Accepted Answer

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

chetan2u · Answer

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

mitrakaushi · Answer

What is the number of pairs here?

According to me it is X(X-3):

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.

LMP · Answer

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.

Avinasht123 · Answer

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 · Answer

Bismark thank you very much for the explanation are you sure you are not an expert

Chetan2u thanks a ton.

Nijssen · Answer

Is there a way to solve this with circular combinations?

SchruteDwight · Answer

First thing you should realize is that 28 minutes = 14*2 minutes.

A circle of 7 is the only one that produces 14 combinations:

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

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

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