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  4. Number of onto (surjective) functions from A to B if n(A) = 6 and n(B) = 3 are

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Q. Number of onto (surjective) functions from $A$ to $B$ if $n(A) = 6$ and $n(B) = 3$ are

Relations and Functions - Part 2

Solution:

Number of onto functions from $A$ to $B$ if $n(A) = m$
$n(B) = n$ and $1 \le n \le m$ are equal to
$\displaystyle\sum_{r = 1}^n (-1)^{n - r} \,{}^nC_r \,r^m$
Here $ n = 3, m = 6$
$\therefore $ Number of onto functions
$ = \displaystyle\sum_{r = 1}^3 (-1)^{3-r} \,\,{}^{3}C_r \,r^6$
$= (-1)^3 \,{}^3C_1 \,1^6 + (-1)^1\,\,{}^3C_2 \,2^6 + (-1)^0 \,{}^3C_3 \,3^6$
$ = 3 - 3 \times 2^6 + 3^6 $
$ = 3(3^5 - 2^6 + 1) = 540$