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If a set $\mathrm{A}$ has $\mathrm{n}$ elements, then the number of functions defined from A to A that are not one-one is
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$\mathrm{n}^{\mathrm{n}}-\mathrm{n} !$
No. of functions from $A$ to $A=n^n$
No. of one-one functions $=\frac{n !}{(n-n) !}=n !$
$\therefore \quad$ No. of functions defined from $A$ to $A$ that are not oneone $=n^n-n !$
No. of one-one functions $=\frac{n !}{(n-n) !}=n !$
$\therefore \quad$ No. of functions defined from $A$ to $A$ that are not oneone $=n^n-n !$
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