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