I have a question regarding the permutation of sets and it is:
Problem: let the sample space $X$ be the set of permutations of $\{1,2,3,4,5\}$, the permutation $\{n_1,n_2,n_3,n_4,n_5\}$ represents the object allocation where for $i,j\in \{1,2,3,4,5\}$ we have $n_i=j$ if person $i$ receives the object by person $j$. Furthermore $i\in \{1,2,3,4,5\}$. If we define the events:
$$A_i=\{(n_1,n_2,n_3,n_4,n_5)\in X\space |\space n_i=i\}$$
My confusions: I do not understand how to list these elements under the defined set of element characteristics this set has for instance, in the sample space $X$, can $n_1=1, n_2=2,n_3=3,n_4=4, n_5=5?$
If not then the values $n_1$, $n_2$,... can take are $n_1=2,3,4,5; n_2=1,3,4,5; ...$ etc. So one possible element of the sample space is $(2,3,4,5,1)\in X?$.
But in the set $A_i,$ now there is a new condition which is $n_i=i$, that means the set $A_1=\{(1,1,1,1,1)\}?$. I am alittle confused on the definition of $n_i=i$ in the set $A_i,$ and how many elements $A_1, A_2,...,A_5$ contains. Can anyone help me explain or find the elements of the set $A_i$, or just an example for$ A_1$ and $A_2?$ I would appreciate it.