How many possible arrangements of 12 people if....

[Clifford]

There are 12 people. How many possible arrangements are there if two particular people must be in the centre?

If the two particular people just have to be together I know you would do it by taking them as one person, so it would be 11! and there are 2! ways to arrange them, so it would be (11!)(2!). But I dont know how to account if they have to be in the center.

Could somebody explain and show me how to do this?

 

[soroban]

Re: Quick Question

Hello, Clifford!

There are 12 people.
How many possible arrangements are there is two particular people
must be in the centre?

If the two particular people just have to be together,
I know you would do it by taking them as one person. Right!

And you also right that there are \(\displaystyle 2!\,=\,2\) ways to order the two people.

Now simply place the pair in the center (in either order):
. . . . _ _ _ _ _ A B _ _ _ _ _

And there are \(\displaystyle 10!\) ways to seat the other ten people.

Answer: \(\displaystyle \:2\,\times\,10!\)