n people standing in a line

Here is the question:

If Sam and Peter are among **n** men who are arranged at random in a line, what is the probability that exactly **k** men stand between them?

My solution:

1. Group Sam and Peter and the **k** men together and count them as one person.

2. So in effect, we have **(n-k-2+1)** number of people.

3. Simplify that and we get **n-k-1**

4. There are **(n-k-1)!** ways of arranging them.

5. There are then **2** more ways of arranging Sam and Peter by switching them around.

6. Finally, there are **k!** ways of arranging the k people between Sam and Peter

7. There are **n!** ways of arranging everyone without any restrictions.

8. Therefore the solution is:

**(n-k-1)! * 2 * k! / n!**

However, the book solution is **2*(n-k-1)/n*(n-1)**

I checked by plugging some numbers and my answer is different from the book. Can someone explain what I'm missing?

Thanks