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Problem: A function $$f$$ from set $$A$$ into set $$B$$ is a rule which assigns each element $$x$$ in $$A$$, a unique (one and only one) element (denoted by $$f(x)$$ in $$B$$. A function of set from $$A$$ into $$B$$ is called an onto function, if for each element $$y$$ in $$B$$ there is some element $$x$$ in $$A$$, such that $$f(x)=y$$. Now suppose that $$A =$$ {$$1,2,\cdots,n$$} and $$B=$${$$1,2,3$$}. Determine the total number of onto functions of $$A$$ into $$B$$.

Solution: For every $$y\in B$$, $$x$$ can take values from the $$n$$ numbers in $$A$$. But they have to be unique, so the first $$y_1$$ can have $$n$$ pre-images, $$y_2$$ can have $$(n-1)$$ pre-images and so on.

Thus in this case, the total number of possible functions is given by : $$n(n-1)(n-2)$$.