1. groups

From a group 2 groups of 2-person groups are formed in 105 ways.How many persons are there in this group.

Answer is 7.But I don't know how.

If we from 2 group of 2 persons from 7 it makes 35.
C(5,1)C(7,1)

2. Re: groups

$\dfrac{\dbinom{n}{2}\dbinom{n-2}{2}}{2}=105$
Simplify:
$\dfrac{n!}{(n-4)!}=840$

Trial and error gives 7

3. Re: groups

I thought different.
There are n persons.If we choose 1 person n-1 remains.
From n-1 we choose 1 person.
C(n-1,1)
n-2 remained. From n-2 we call one person and put him aside.
n-3 remained.From that we choose 1.
And
C(n-1,1)C(n-3,1)

I thought that way. I think it is wrong.

You divided by 2 because there are 2 groups. We don't want c,a and a,c at the same time. Am I right?

4. Re: groups

Originally Posted by kastamonu
I thought different.
There are n persons.If we choose 1 person n-1 remains.
From n-1 we choose 1 person.
C(n-1,1)
n-2 remained. From n-2 we call one person and put him aside.
n-3 remained.From that we choose 1.
And
C(n-1,1)C(n-3,1)

I thought that way. I think it is wrong.

You divided by 2 because there are 2 groups. We don't want c,a and a,c at the same time. Am I right?
There are two indistinguishable groups. If we choose a,b for the first group and c,d for the second, that's the same as c,d for the first group and a,b for the second.

Many Thanks.