Fix point (arithmetic mean)

WE mark:

* For every function f:A-->A we mark fix(f):={a in A|f(a)=a} all sets of fix points of f.

* In S_n we mark all the sets of permutations on {1,2,3,...,n}

* In T_n we mark all functions f:{1,2,3,...,n}-->{1,2,3,...,n}.

Questions:

1. How many fix points in arithmetic mean there is to function f:{1,2,3,...,n}-->{1,2,3,...,n}?

In other words, compute the {sigma{ f in T_n, |fix(f)|}}/|T_n|

a. Compute top of fraction with changing order of sum.

b. Compute top of fraction with using Newton's binomial formula.

2. Compute:

sigma{( f in S_n, |fix(f) ) CHOOSE 2}