Dividing students into teams-combinatorics In how many ways can $n$ number of students be divided into two teams such that each team has at least one student.  
This is what I did:
Let $x_1$ be the number of students in the one team and $x_2$ be the number of students in the other team.Then number of ways of dividing students is equivalent to
$x_1+x_2=n$ such that $x_1,x_2>=1$.
I let $y_1=x_1-1$ and $y_2=x_2-1$.
Then the number of ways of dividing students are $n-1 \choose 1 $=$n-1$  
The given answer in the book is $2^{n-1}-1$.
How is this answer obtained.Can someone please show me what's wrong with what I did.

One of the students is Alphonse.  Let $N$ be the set of non-Alphonses. We need to choose a subset of $N$ to be on Alphonse's team, but we cannot choose all of $N$.  The set $N$ has $2^{n-1}-1$ subsets other than $N$.
