Question

Find the number of all onto functions from the set {1, 2, 3, … , n) to itself.

Answer 1

$\textbf{Onto function from the set \{1, 2, 3, ..., n\}}\\\textbf{ to itself is simply a permutation on }\\\textbf{n symbols 1, 2, 3, …, n.}$

Therefore, the total number of onto maps from {1, 2, 3, …, n} to itself is the same as the total number of permutations on n symbols 1, 2, 3, …, n, which is $\bf{n!}$

Answer 2

Hey !

Onto function: - A function f : A ---> B is said to be a onto function or surjection if every element of A if f(A) = B or range of f is the co-domain of f.

So, f: A ---> B is Surjection for each b ∈ B, there exists a ∈ B such that f(a) = b.

Now, f : A → A where A = {1 , 2, 3,.....,n}

All onto function

It's a permutation of n symbols 1,2,3,....,n

Thus,

Total number of Onto maps from A = {1 , 2, 3,....,n}

Total number of permutations of n symbols 1,2,3,......,n

GOOD LUCK !