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

Question

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

This is a  Data Sufficiency Butler Question

Check the links to other Butler Projects: 
 
   Data Sufficiency Butler 
   Problem Solving Butler  
   Sentence Correction Butler  
   Reading Comprehension Butler  
   Integrated Reasoning Butler

kjs10927 · Accepted Answer

Answer is C.

2 people = 1 clinks
3 people = 3 clinks
4 people = 6 clinks
5 people = 10 clinks
6 people = 15 clinks
7 people = 21 clinks
8 people = 28 clinks
9 people = 36 clinks
10 people = 45 clinks
11 people = 55 clinks
12 people = 66 clinks.

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

If current # of people is 10, then 2 fewer would be 8. Corresponding # of clinks is 45 and 28. 45 - 28 = 17 (O)
If current # of people is 11, then 2 fewer would be 9. Corresponding # of clinks is 55 and 36. 55 - 36 = 19 (O)

There could be 10 or 11 people.

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

If current # of people is 9 then 2 more would be 11. Corresponding # of clinks is 36 and 55. 55 - 36 = 19 (O)
If current # of people is 10, then 2 more would be 12. Corresponding # of clinks is 45 and 66. 66 - 45 = 21 (O)

There could be 9 or 10 people.

Together, we know 10 people is the answer.

ribbons · Answer

Is there a typo ? because both statements are about two FEWER ppl

Bunuel · Answer

_______________
Fixed. Thank you!

Nishumachete12 · Answer

Answer C. 10 people.

Purnank · Answer

Given - 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. each clink is produced by the glasses of only two people.
Assuming n people seated at the table then it is like selecting 2 from group of n peoples = nC2
To calculate how many people are seated at the table lets check given conditions.

1st - If  two fewer people were seated  at the table, there would be at least 17 but no more than 19 fewer clinks.
Therefore,
17 <= (n-2)C2 <= 19
lets check by substituting values n = 2, 3, 4, 5, 6, and so on. Therefore for n=8, n-2=6, 6c2 = 15 and for n=9, n-2=7, 7c2 = 21.
Therefore you will see no such value exist hence not sufficient.

2nd - If  two more people were seated  at the table, there would be at least 18 but no more than 21 fewer clinks.
18 <= (n+2)C2 <= 21 now if we put n=5, n+2 = 7, 7c2=21. Which is under given limit and the only possible value hence sufficient.

Answer B.

vj07 · Answer

is the answer B? i am not getting a specific ans for st 1

ribbons · Answer

Bunuel   chetan2u   KarishmaB  or any other expert

Please chime in, here the fewer or more clinks refers to total number of clinks or the increase in clinks

vj07 · Answer

understood i overlooked the "fewer" part in the statements somehow

chetan2u · Answer

Fewer or more links here mean the increase or decrease in number of clinks.

Posted from my mobile device

chetan2u · Answer

Let n people are there, and nC2 is the number of clinks, that is the number of ways to choose two out of n people.

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 number of people is (n-2) and number of clinks = (n-2)C2
Thus, nC2-(n-2)C2 is 17, 18 or 19.
 \frac{n(n-1)}{2} - \frac{(n-2)(n-3)}{2} = \frac{n^2-n+n^2+5n-6}{2} = 2n-3 
 2n-3=17…n=10 
 2n-3=18 …n cannot be fraction
 2n-3=19…n=11 
Insufficient

(2) If two more people were seated at the table, there would be at least 18 but no more than 21 fewer clinks.
Now, the number of people is (n+2) and number of clinks = (n+2)C2
Thus, (n+2)C2 is 18, 19, 20 or 21
 \frac{(n+2)(n+1)}{2} - \frac{(n-1)(n)}{2} = \frac{n^2+3n+2-n^2-n}{2} = 2n+1 
 2n+1=18 … n cannot be fraction
 2n+1=19…n=9 
 2n+1=20 … n cannot be fraction
 2n+1=21…n=10 
Insufficient

Combined
n=10
sufficient
C

Posted from my mobile device

delacream1789 · Answer

Didnot understand why are we substracting ?

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

Prep Toolkit

Top 5 GMAT Debrief Videos

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

GMAT Data Sufficiency (DS) Questions

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

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