In how many different ways can the $10$ people be talking? 
(a) There are $10$ people at a party who form pairs to talk to each other, with everyone talking to someone else (so $5$ pairs in total).
  In how many different ways can the $10$ people be talking?
(b) Suppose now that there are $11$ people at the party, so that now the guests form $5$ pairs and $1$ person is left by themselves.
  How many configurations are possible now?
(Original picture of the above text here.)

I've got the idea on how to do (a) where I got $945$ as my answer from $\frac{10!}{(2^5) \cdot 5!}$.
For (b), I solved it by $10 \cdot 8 \cdot 6 \cdot 4 \cdot 2 = 3840$.
Can someone tell me if my answers are right for both (a) and (b). 
Thanks
 A: 
There are $10$ people at a party who form pairs to talk to each other, with everyone talking to someone else (so $5$ pairs total).  In how many different ways can the $10$ people be talking?

Your answer is correct.
Here is another method that confirms your result.
Line up the $10$ people in some order, say by age.  The first person in line can choose a conversation partner in $9$ ways.  Remove those two people from the line, which leaves us with eight people in the line.  The first person remaining in the line can choose a conversation partner in $7$ ways.  Remove those two people from the line, which leaves us with six people in the line.  The first person remaining in line can choose a conversation partner in $5$ ways.  Remove those two people from the line, which leaves us with four people in the line.  The first person remaining in line can choose a conversation partner in $3$ ways.  Remove those two people from the line, which leaves us with two people.  They must be conversation partners. Hence, the number of possible arrangements is 
$$9 \cdot 7 \cdot 5 \cdot 3 \cdot 1 = 945$$
which can be expressed in the form $9!!$.  The symbol $!!$ is read double factorial.

Suppose now that there are eleven people at the party, so that the guests form $5$ pairs and $1$ person is left by themselves.  How many configurations are possible now.

Your answer is incorrect since you have not chosen the person who does not have a conversation partner.
We choose which person does not have a conversation partner.  That leaves $10$ people who form pairs to talk to each other.  Thus, the answer is $11$ times the result above.
$$11 \cdot 9 \cdot 7 \cdot 5 \cdot 3 \cdot 1 = 10,395$$
