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

Solution:

f: X → Y is said to be onto, if every element of Y is the image of some element of X under f, i.e, for every y ϵ Y,

there exists an element a in X such that f(x) = y

Onto functions from the set {1, 2, 3, ...., n}

to itself is simply a permutation on n symbols 1, 2, 3, ...., n.

Thus,

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 n!

NCERT Solutions for Class 12 Maths - Chapter 1 Exercise ME Question 10

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

Summary:

The number of all onto functions from the set {1, 2, 3, .... n} to itself is n!