We define the sample space $\Omega:=\{\omega\in\{1,\dots n\}^n\mid \omega_i\neq\omega_j\text{ for all } 1\leq i<j\leq n\}$. We call a tupel $\omega\in\Omega$ fixed point free if for all $1\leq i\leq n:$ $\omega_i\neq i$. How many tupels exist that have exactly one fixpoint?
Our sample solution says:
$n\cdot D_{n-1,0}$, where $D_{n-1,0}$ is the respective Rencontre number in the setting of $n-1$ tupels and $0$ fixed points. (see https://en.wikipedia.org/wiki/Rencontres_numbers)
We can choose $n$ fixed points and for the remaining $n-1$ entries there are $D_{n-1,0}$ possibilities of permutations without fixed points.
What I don't understand is: Why are we allowed to establish a connection between the $n$-tupels and the $n-1$ tupels?
Intuitively it makes sense but it doesn't feel rigorous. Don't we have to construct a bijection between fixed point free tupels of lenght $n-1$ and tupels of length $n$ that have exactly one fixed point? If so, how do we construct such a bijection?