From the collection of all permutation matrices of size $10\times10$, one such matrix is randomly picked. What is the expected value of its trace? (A permutation matrix is one that has precisely one non-zero entry in each column and in each row, that non-zero entry being 1.)
I know that possible options for traces are $0,1,2,3,4,5,6,7,8,9,10$. Now from this how to find the expectation of the trace?
Let $A_{ij}$ denote row $i$, column $j$, of matrix $A$.
Let $G$ be the set of $10\times10$ permutation matrices. Then the expected trace is
$$\begin{align*} \frac{1}{10!}\sum_{A\in G}\text{tr}(A) &= \frac{1}{10!}\sum_{A\in G}\sum_{i=1}^{10}A_{ii} \\ &= \frac{1}{10!}\sum_{i=1}^{10}\sum_{A\in G}A_{ii} \\ &= \frac{1}{10!}\sum_{i=1}^{10}9! \\ &= \frac{10\cdot9!}{10!} \\ &= 1 \end{align*}$$
Note that $\underset{A\in G}{\sum}A_{ii}=9!$, because each permutation matrix $A$ has $A_{ii}=0$ or $A_{ii}=1$. The ones with $A_{ii}=1$ are the ones that correspond to the permutations which send $i\mapsto i$, and there are $9!$ of those.