  We prove bijectively that the total number of cycles of all even permutations
of $[n]=\{1,2,...,n\}$ and the total number of cycles of all odd permutations
of $[n]$ differ by $(-1)^n(n-2)!$, which was stated as an open problem by
Mikl\'{o}s B\'{o}na. We also prove bijectively the following more general
identity: $$\sum_{i=1}^n c(n,i)\cdot i \cdot (-k)^{i-1} = (-1)^k k! (n-k-1)!,$$
where $c(n,i)$ denotes the number of permutations of $[n]$ with $i$ cycles.
