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

Explanation

Let take set `{1,2,3}`

As f is onto,all elements of`{1,2,3}` have unique pre-image.

Hence total no. of one-one function=`3×2×1=6`

Let f is onto ,then all elements of `{1,2,3}` have unique pre-image.

Hence total no. of onto functions=`n×n−1×n−2×2×1=n!`

Hence, the number of all
onto functions from the set `A={1,\ 2,\ 3,......\ n}`
to itself=`n!`