# 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?

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## 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

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 o

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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

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## N persons stand on the circumference of a circle at distinct points. : Problem Solving (PS)

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 ...

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

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**Bunuel**

Math Expert

Joined: **02 Sep 2009 **

Posts: **86747**

N persons stand on the circumference of a circle at distinct points. [#permalink]

12 Mar 2021, 02:16 3 Bookmarks Expert Reply 00:00 A B C D E

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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) 7 (B) 6 (C) 5 (D) 4 (E) 3 Show Answer _________________

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**chetan2u**

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Re: N persons stand on the circumference of a circle at distinct points. [#permalink]

12 Mar 2021, 18:27 1 Bookmarks Expert Reply Bunuel wrote:

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) 7 (B) 6 (C) 5 (D) 4 (E) 3

Total number of ways to choose a pair = NC2

Number of pairs in which both the persons are standing next to each other =N

Ways in which the persons are not next to each other =NC2-N

However it will also be equal to 28/2, as each possible pair sings for 2 minutes.

Thus, N(N−1) 2 −N= 28 2 N(N−1)2−N=282 N(N−1)−N=28 N(N−1)−N=28 N 2 −3N−28=0 N2−3N−28=0 (N−7)(N+4)=0 (N−7)(N+4)=0

So N can be 7 or -4. As it cannot be negative, N=7.

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Re: N persons stand on the circumference of a circle at distinct points. [#permalink]

23 Mar 2021, 04:43 1 Kudos Expert Reply Bunuel wrote:

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) 7 (B) 6 (C) 5 (D) 4 (E) 3 Solution:

Assume the N people are vertices of an N-gon. Then all possible pairs of people that can be formed without any pair standing next to each other is also the number of diagonals of the N-gon. Recall that the number of diagonals of an N-gon is N(N - 3) / 2, and we see that a total of 28/2 = 14 songs are sung; therefore, we can create the equation:

N(N - 3)/2 = 14 N(N - 3) = 28

Without solving the equation algebraically, we see that N must be 7 since 7(7 - 3) = 7(4) = 28.

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Re: N persons stand on the circumference of a circle at distinct points. [#permalink]

23 Mar 2021, 04:43 NEW TOPIC

### Similar topics

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

AUTHOR Bunuel 13 Dec 2020, 05:26 7 Replies 5 ER

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

AUTHOR LMP

## 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

Answer (1 of 3): for a person no of such pair = N-3 ( excluding himself, and 2 of it adjacent persons) so total no. of pair for N person = N×(N-3) but here each pair has been counted twice so actual no. of pair = N×(N-3)/2 so total singing time = N×(N-3)/2×2 = N×(N-3) so N×(N-3) = 28 so N = 7

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?

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for a person no of such pair = N-3 ( excluding himself, and 2 of it adjacent persons)

so total no. of pair for N person = N×(N-3)

but here each pair has been counted twice so actual no. of pair = N×(N-3)/2

so total singing time = N×(N-3)/2×2 = N×(N-3)

so N×(N-3) = 28 so N = 7 Related questions

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Munish Fauzdar

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Clearly there are 14 songs which got sung in the conference, meaning which there are possible 14 such pairs which can be formed picking two when doesn’t include adjacent pairs.

Let n be the number of people standing in a circle.

Thought: The steps will be that if we take two people out of n, there are nC2 selections possible. However from these selections there will be n selections which would be reduced due to being formed with the adjacent person.

Hence, if total songs played were for 28 mins, => 14 songs were played => 14 pairs were formed.

n C 2 - n = 14 n (n-1)/2 - n = 14 n^2 - n - 2n = 28 n^2 -

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Tushar Kaviti

B. Tech in Electronics and Communication Engineering, JNTUK University College of Engineering Narsaraopet (Graduated 2020)Author has 72 answers and 50.8K answer views2y

28 min makes 14 pairs. now the persons are standing in circle and persons not standing next to each other are forming 14 pairs. imagine connecting the pairs and you see polygons. the pairs not adjacent to each other form 14 pairs. this is like counting the no of diagonals because diagonals are formed between non adjacent points except here we have persons as points here. noe diagonals should be 14

so nc2-n= 14 which gives us no of sides as 7.

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Andrew Winkler

PhD in Mathematics, Courant Institute of Mathematical Sciences, NYU (Graduated 1987)Upvoted by

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Two points on the circumference of a circle of radius √2 are chosen 1 at a time and at random. From each, left or right is randomly chosen and then a chord of length 2 is drawn in this direction. What's the probability that the two chords intersect?

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