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A group of people is seated at a table at which a toast is made. Follo

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A group of people is seated at a table at which a toast is made. Follo [#permalink] New post   06 Dec 2018, 17:43
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A group of people is seated at a table at which a toast is made. Following the toast, each person must clink glasses exactly once with each of the other people at the table. If each clink is produced by the glasses of only two people, how many people are seated at the table?

1. If two fewer people were seated at the table, there would be at least 17 but no more than 19 fewer clinks.

2. If two fewer people were seated at the table, there would be at least 18 but no more than 21 fewer clinks.
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A group of people is seated at a table at which a toast is made. Follo [#permalink] New post   06 Dec 2018, 22:35
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Graina wrote:
A group of people is seated at a table at which a toast is made. Following the toast, each person must clink glasses exactly once with each of the other people at the table. If each clink is produced by the glasses of only two people, how many people are seated at the table?

1. If two fewer people were seated at the table, there would be at least 17 but no more than 19 fewer clinks.

2. If two fewer people were seated at the table, there would be at least 18 but no more than 21 fewer clinks.


let there be n people, so basically we are asked to find 2 out of n people.
nC2..

let us see each choice now...
1. If two fewer people were seated at the table, there would be at least 17 but no more than 19 fewer clinks.
Now, the strength becomes n-2, so ways you can choose 2 out of n-2 = (n-2)C2
So, \(17\leq{nC2-(n-2)C2}\leq{19}\).
let us find (n)C2-(n-2)C2 => \(\frac{n!}{(n-2)!2!}-\frac{(n-2)!}{(n-2-2)!2!}=\frac{n(n-1)}{2}-\frac{(n-2)(n-3)}{2}=\frac{n^2-n-n^2+5n-6}{2}=2n-3\)..
so \(17\leq{2n-3}\leq{19}=20\leq{2n}\leq{22}\)=\(10\leq{n}\leq{11}\)..
so n can be 10 or 11
insuff..


2. If two fewer people were seated at the table, there would be at least 18 but no more than 21 fewer clinks.
Now, the strength becomes n-2, so ways you can choose 2 out of n-2 = (n-2)C2
So, \(18\leq{nC2-(n-2)C2}\leq{21}\).
let us find nC2-(n-2)C2 => \(\frac{n!}{(n-2)!2!}-\frac{(n-2)!}{(n-2-2)!2!}=\frac{n(n-1)}{2}-\frac{(n-2)(n-3)}{2}=\frac{n^2-n-n^2+5n-6}{2}=2n-3\)..
so \(18\leq{2n-3}\leq{21}=21\leq{2n}\leq{24}\)=\(10.5\leq{n}\leq{12}\)..
so n can be 11 or 12
insuff..

Combined..
n can be 11..
sufficient..

C


You can also work with taking different values of n..
find 8C2-6C2 and then 9C2-7C2 and so on..
match with your statement and you will get your answer..
example 9C2-7C2 = 9*4-7*3=36-21=15, so increase n by 1..
10C2-8C2=5*9-4*7=45-28=17..
11C2-9C2=11*5-9*4=55-36=19..
so n as 10 and 11 are correct.

As we can see with increase of 1 in n, the possible value of n increases by 2..
so, if 10C2-8C2=17, 11C2-9C2=19, 12C2-11C2=21..
so statement I gives us 17 and 19 and statement II gives us 19 and 21..
Thus, 19 is the answer..
C
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Re: A group of people is seated at a table at which a toast is made. Follo [#permalink] New post   07 Dec 2018, 22:29
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ritu1009 wrote:
chetan2u wrote:
Graina wrote:
A group of people is seated at a table at which a toast is made. Following the toast, each person must clink glasses exactly once with each of the other people at the table. If each clink is produced by the glasses of only two people, how many people are seated at the table?

1. If two fewer people were seated at the table, there would be at least 17 but no more than 19 fewer clinks.

2. If two fewer people were seated at the table, there would be at least 18 but no more than 21 fewer clinks.


let there be n people, so basically we are asked to find 2 out of n people.
nC2..

let us see each choice now...
1. If two fewer people were seated at the table, there would be at least 17 but no more than 19 fewer clinks.
Now, the strength becomes n-2, so ways you can choose 2 out of n-2 = (n-2)C2
So, \(17\leq{(n-2)C2-nC2}\leq{19}\).
let us find (n-2)C2-nC2 => \(\frac{n!}{(n-2)!2!}-\frac{(n-2)!}{(n-2-2)!2!}=\frac{n(n-1)}{2}-\frac{(n-2)(n-3)}{2}=\frac{n^2-n-n^2+5n-6}{2}=2n-3\)..
so \(17\leq{2n-3}\leq{19}=20\leq{2n}\leq{22}\)=\(10\leq{n}\leq{11}\)..
so n can be 10 or 11
insuff..


2. If two fewer people were seated at the table, there would be at least 18 but no more than 21 fewer clinks.
Now, the strength becomes n-2, so ways you can choose 2 out of n-2 = (n-2)C2
So, \(18\leq{(n-2)C2-nC2}\leq{21}\).
let us find (n-2)C2-nC2 => \(\frac{n!}{(n-2)!2!}-\frac{(n-2)!}{(n-2-2)!2!}=\frac{n(n-1)}{2}-\frac{(n-2)(n-3)}{2}=\frac{n^2-n-n^2+5n-6}{2}=2n-3\)..
so \(18\leq{2n-3}\leq{21}=21\leq{2n}\leq{24}\)=\(10.5\leq{n}\leq{12}\)..
so n can be 11 or 12
insuff..

Combined..
n can be 11..
sufficient..

C


You can also work with taking different values of n..
find 8C2-6C2 and then 9C2-7C2 and so on..
match with your statement and you will get your answer..
example 9C2-7C2 = 9*4-7*3=36-21=15, so increase n by 1..
10C2-8C2=5*9-4*7=45-28=17..
11C2-9C2=11*5-9*4=55-36=19..
so n as 10 and 11 are correct.

As we can see with increase of 1 in n, the possible value of n increases by 2..
so, if 10C2-8C2=17, 11C2-9C2=19, 12C2-11C2=21..
so statement I gives us 17 and 19 and statement II gives us 19 and 21..
Thus, 19 is the answer..
C


can u please explain this equation \(17\leq{(n-2)C2-nC2}\leq{19}\).
why its (n-2)C2-nC2?



Hello

If there are total 'n' people, number of clinks produced will be nC2.
If there are total 'n-2' people, number of clinks produced will be (n-2)C2.

First statement says that in case of 2 fewer people, the 'difference in number of clinks produced' will be between 17 and 19 inclusive.
And what is the difference in number of clinks produced ? It will be nC2 - (n-2)C2 only.

Thats why Chetan has made the equation:
17 <= [nC2 - (n-2)C2] <= 19

(I think just by typo Chetan has written (n-2)C2 - nC2.. instead he should have written nC2 - (n-2)C2.
But his calculation is completely fine, and all else that he has written is also perfect).
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Re: A group of people is seated at a table at which a toast is made. Follo [#permalink] New post   06 Dec 2018, 23:45
Graina wrote:
A group of people is seated at a table at which a toast is made. Following the toast, each person must clink glasses exactly once with each of the other people at the table. If each clink is produced by the glasses of only two people, how many people are seated at the table?

1. If two fewer people were seated at the table, there would be at least 17 but no more than 19 fewer clinks.

2. If two fewer people were seated at the table, there would be at least 18 but no more than 21 fewer clinks.





Can some one please explain, "each person must clink glasses exactly once with each of the other people at the table. If each clink is produced by the glasses of only two people" ?
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Re: A group of people is seated at a table at which a toast is made. Follo [#permalink] New post   07 Dec 2018, 00:54
jashandeep2332 wrote:
Graina wrote:
A group of people is seated at a table at which a toast is made. Following the toast, each person must clink glasses exactly once with each of the other people at the table. If each clink is produced by the glasses of only two people, how many people are seated at the table?

1. If two fewer people were seated at the table, there would be at least 17 but no more than 19 fewer clinks.

2. If two fewer people were seated at the table, there would be at least 18 but no more than 21 fewer clinks.





Can some one please explain, "each person must clink glasses exactly once with each of the other people at the table. If each clink is produced by the glasses of only two people" ?



to explain above statement , for example take 5 persons seated and named as a,b,c,d,e

first person a can clink with b,c,d,e .so on total there will be 4 clinks
in the same way, second person can clink with b,c,e on total there will be 3 clinks... so on

Hope answered your question
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Re: A group of people is seated at a table at which a toast is made. Follo [#permalink] New post   07 Dec 2018, 02:27
chetan2u wrote:
Graina wrote:
A group of people is seated at a table at which a toast is made. Following the toast, each person must clink glasses exactly once with each of the other people at the table. If each clink is produced by the glasses of only two people, how many people are seated at the table?

1. If two fewer people were seated at the table, there would be at least 17 but no more than 19 fewer clinks.

2. If two fewer people were seated at the table, there would be at least 18 but no more than 21 fewer clinks.


let there be n people, so basically we are asked to find 2 out of n people.
nC2..

let us see each choice now...
1. If two fewer people were seated at the table, there would be at least 17 but no more than 19 fewer clinks.
Now, the strength becomes n-2, so ways you can choose 2 out of n-2 = (n-2)C2
So, \(17\leq{(n-2)C2-nC2}\leq{19}\).
let us find (n-2)C2-nC2 => \(\frac{n!}{(n-2)!2!}-\frac{(n-2)!}{(n-2-2)!2!}=\frac{n(n-1)}{2}-\frac{(n-2)(n-3)}{2}=\frac{n^2-n-n^2+5n-6}{2}=2n-3\)..
so \(17\leq{2n-3}\leq{19}=20\leq{2n}\leq{22}\)=\(10\leq{n}\leq{11}\)..
so n can be 10 or 11
insuff..


2. If two fewer people were seated at the table, there would be at least 18 but no more than 21 fewer clinks.
Now, the strength becomes n-2, so ways you can choose 2 out of n-2 = (n-2)C2
So, \(18\leq{(n-2)C2-nC2}\leq{21}\).
let us find (n-2)C2-nC2 => \(\frac{n!}{(n-2)!2!}-\frac{(n-2)!}{(n-2-2)!2!}=\frac{n(n-1)}{2}-\frac{(n-2)(n-3)}{2}=\frac{n^2-n-n^2+5n-6}{2}=2n-3\)..
so \(18\leq{2n-3}\leq{21}=21\leq{2n}\leq{24}\)=\(10.5\leq{n}\leq{12}\)..
so n can be 11 or 12
insuff..

Combined..
n can be 11..
sufficient..

C


You can also work with taking different values of n..
find 8C2-6C2 and then 9C2-7C2 and so on..
match with your statement and you will get your answer..
example 9C2-7C2 = 9*4-7*3=36-21=15, so increase n by 1..
10C2-8C2=5*9-4*7=45-28=17..
11C2-9C2=11*5-9*4=55-36=19..
so n as 10 and 11 are correct.

As we can see with increase of 1 in n, the possible value of n increases by 2..
so, if 10C2-8C2=17, 11C2-9C2=19, 12C2-11C2=21..
so statement I gives us 17 and 19 and statement II gives us 19 and 21..
Thus, 19 is the answer..
C


can u please explain this equation \(17\leq{(n-2)C2-nC2}\leq{19}\).
why its (n-2)C2-nC2?
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Re: A group of people is seated at a table at which a toast is made. Follo [#permalink] New post   12 Dec 2018, 02:53
This was my thought process without the need of any calculations

-- clinks are made only once by each member, 1 clink requires two persons. Once clinked members are out of scope.

1. If two fewer people were seated at the table, there would be at least 17 but no more than 19 fewer clinks.

--let the current people be n+2,
--now statement 1 says
17<=n/2<=19
=> 34<=n<=38
=> 34<= current_people <=38 ----------------------II

Thus current people = n+2
=> 36<= n+2 <= 40



2. If two fewer people were seated at the table, there would be at least 18 but no more than 21 fewer clinks.
--let the current people be n+2,
--now statement 1 says
18<= n/2 <=21
=> 36<= n <=42

Thus current people = n+2
=> 38<= n+2 <= 44
=> 38<= current_people <= 44 ----------------- I

Combining I and II

current_people = 38
Hence C
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Re: A group of people is seated at a table at which a toast is made. Follo [#permalink] New post   12 Dec 2018, 03:14
chetan2u wrote:
Graina wrote:
A group of people is seated at a table at which a toast is made. Following the toast, each person must clink glasses exactly once with each of the other people at the table. If each clink is produced by the glasses of only two people, how many people are seated at the table?

1. If two fewer people were seated at the table, there would be at least 17 but no more than 19 fewer clinks.

2. If two fewer people were seated at the table, there would be at least 18 but no more than 21 fewer clinks.


let there be n people, so basically we are asked to find 2 out of n people.
nC2..

let us see each choice now...
1. If two fewer people were seated at the table, there would be at least 17 but no more than 19 fewer clinks.
Now, the strength becomes n-2, so ways you can choose 2 out of n-2 = (n-2)C2
So, \(17\leq{nC2-(n-2)C2}\leq{19}\).
let us find (n)C2-(n-2)C2 => \(\frac{n!}{(n-2)!2!}-\frac{(n-2)!}{(n-2-2)!2!}=\frac{n(n-1)}{2}-\frac{(n-2)(n-3)}{2}=\frac{n^2-n-n^2+5n-6}{2}=2n-3\)..
so \(17\leq{2n-3}\leq{19}=20\leq{2n}\leq{22}\)=\(10\leq{n}\leq{11}\)..
so n can be 10 or 11
insuff..


2. If two fewer people were seated at the table, there would be at least 18 but no more than 21 fewer clinks.
Now, the strength becomes n-2, so ways you can choose 2 out of n-2 = (n-2)C2
So, \(18\leq{nC2-(n-2)C2}\leq{21}\).
let us find nC2-(n-2)C2 => \(\frac{n!}{(n-2)!2!}-\frac{(n-2)!}{(n-2-2)!2!}=\frac{n(n-1)}{2}-\frac{(n-2)(n-3)}{2}=\frac{n^2-n-n^2+5n-6}{2}=2n-3\)..
so \(18\leq{2n-3}\leq{21}=21\leq{2n}\leq{24}\)=\(10.5\leq{n}\leq{12}\)..
so n can be 11 or 12
insuff..

Combined..
n can be 11..
sufficient..

C


You can also work with taking different values of n..
find 8C2-6C2 and then 9C2-7C2 and so on..
match with your statement and you will get your answer..
example 9C2-7C2 = 9*4-7*3=36-21=15, so increase n by 1..
10C2-8C2=5*9-4*7=45-28=17..
11C2-9C2=11*5-9*4=55-36=19..
so n as 10 and 11 are correct.

As we can see with increase of 1 in n, the possible value of n increases by 2..
so, if 10C2-8C2=17, 11C2-9C2=19, 12C2-11C2=21..
so statement I gives us 17 and 19 and statement II gives us 19 and 21..
Thus, 19 is the answer..
C




HI chetan2u
1. If two fewer people were seated at the table, there would be at least 17 but no more than 19 fewer clinks.

I cannot understand this statement.
According to me, it means:
if there are n people
then it becomes n-2

So 17<n-2c2<19
if there are 2 fewer people seated at the table, there would min 17 and max of 19.
Why are you saying that th DIFFERENCE will be min 17 and max 19.
you could simply also say that with 2 fewer people, the clinks shall be min 17 and max 19.

Please tell me if I am going wrong t=somewhere.

Regards
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Re: A group of people is seated at a table at which a toast is made. Follo [#permalink] New post   08 Mar 2020, 16:45
GaganSingh0603 wrote:
A group of people is seated at a table at which a toast is made. Following the toast, each person must clink glasses exactly once with each of the other people at the table. If each clink is produced by the glasses of only two people, how many people are seated at the table?

1. If two fewer people were seated at the table, there would be at least 17 but no more than 19 fewer clinks.

2. If two fewer people were seated at the table, there would be at least 18 but no more than 21 fewer clinks.


So, it talks about it needs two to make a toast. If there are 10 persons, there will be 5 toast. If there is 11 person, still 5 toasts will be made. From this understanding, we can explain the two statements in the following way:

1) Two fewer people would have made the number of persons in that table at least 34(17*2) and at best 39 (19*2 + 1). If there were 37 people actually in the table, then 35 people will make 17 clinks. Again, if there were 40 people in the original setting, 2 fewer people would have made 19 clinks. No single answer. Insufficient.

2) Same as 1, the original no of people can be from 38 to 44 people. No single value can be found. Not possible.

Together, if 2 fewer people were there, the clinks will be no fewer than 18 and no more than 19. So the number of people has to be two more than it requires to make 18- 19 clinks ( 36 - 39).So the number of people can be from 38 - 41. No single value cannot be drawn. Hence, insufficient.

E is the answer.
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Re: A group of people is seated at a table at which a toast is made. Follo [#permalink] New post   15 Dec 2021, 10:02
chetan2u

Are we expected to simplify such elaborate inequalities under 2 minutes ? Also is this a realistic GMAT problem? Just asking because of the lengthy calculation and the brute force that needs to be put in a question of a timed test.
Also is there a conceptual way of arriving at the answer ?
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Re: A group of people is seated at a table at which a toast is made. Follo [#permalink]
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