Question

Let set A={1,2,3,...,n}, and let set X be the collection of all the functions from A to A. We randomly pick a function f from X. (1). What is the probability that f is injective?

Answer 1

`(n1)/(n^n)`

Explanation:

Given that set A={1,2,3,...,n},

and set X be the collection of all the functions from A to A.

From A to A functions should be such that each element in A has a unique image in A

Each element has n choices to select the image

So total number of functions from A to A=
`n*n*...ntimes\\=n^n`

If injective if one element in A is selected as image it should not be image for other element

So first element has n ways, second n-1 ways and so on

No .of injective functions =

`n(n-1)....1\\=n!`

So probability for injective funcitons

=
`(n1)/(n^n)`