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

Answer:

$\frac{n1}{n^n}$

Step-by-step 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

=$\frac{n1}{n^n}$