11. Let $S=\{1,2,3, \cdots \cdots n\}$, and $A$ be a subset of $S$. Arrange the elements of $A$ in descending order, then alternately subtract or add the subsequent numbers starting from the largest number to get the alternating sum of $A$. For example, if $A=\{1,4,9,6,2\}$, rearranging it gives $\{9,6,4,2,1\}$, and its alternating sum is $9-6+4-2+1=9$. Find the sum of all such alternating sums.
11. Let the set of all subsets of $S$ that contain $n$ be denoted as $M$, and the set of all subsets of $S$ that do not contain $n$ be denoted as $N$. Define $f:\left\{n, a_{1}, a_{2}, \cdots \cdots, a_{r}\right\} \rightarrow\left\{a_{1}, a_{2}, \cdots a_{r}\right\}$, which can form a 1-1 mapping from $M$ to $N$. The alternating sum of $\left\{n, a_{1}, a_{2}, \cdots, a_{r}\right\}$ and $\left\{a_{1}, a_{2}, \cdots, a_{r}\right\}$ is $n$. Since the number of subsets of $S$ that contain $n$ is $2^{n-1}$, i.e., $|M|=2^{n-1}$, the sum of the alternating sums of all subsets of $S$ is $n \cdot 2^{n-1}$.