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

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Solution

Taking set ${1, 2, 3}$
Since $f$ is onto,all elements of ${1, 2, 3}$ have unique pre-image.
Total no. of one-one function $= 3 \times 2 \times 1 = 6$
eg:-Since $f$ is onto,all elements of ${1, 2, 3}$ have unique pre-image.
total no. of onto functions $= n \times n - 1 \times n - 2 \times 2 \times 1 = n!$

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

Taking set {1,2,3}Since f is onto,all elements of{1,2,3} have unique pre-image.Total no. of one-one function=3×2×1=6eg:-Since f is onto,all elements of {1,2,3} have unique pre-image.total no. of onto functions=n×n−1×n−2×2×1=n!

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