Q. 103.8( 17 Votes )

# Find the number of all onto functions from the set A = {1, 2, 3, …., n} to itself.

Answer :

**TIP:** –

__Onto Function__: – A function is said to be a onto function or surjection if every element of A i.e, if f(A) = B or range of f is the co – domain of f.

So, is Surjection iff for each , there exists 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} to itself =

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

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