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If $A$ and $B$ are two sets that consist of $m$ and $n$ elements respectively $(n>m)$, then how many one-one function can be defined from $A$ to $B$ ? (I am able to understand the question but not able to come up with an expression, the answer given by my teacher was that it is $C(n,m)\cdot m!$ or $P(n,m)$ (they're the same thing)).

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2 Answers 2

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One to one means two or more elements in set $A$ can't be mapped to a single element in set $B$. So each element in $A$ is to be mapped to a unique element in $B$. This means $m$ out of $n$ elements in $B$ need to be selected for mapping. This can be done in $C(n,m)$ ways. Once this is done, it is a simple case of permuting, where the first element in $A$ has $m$ options in $B$, second has $(m-1)$ and so on.

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HINT:

Think of $A$s elements lined up in a row. Choose $m=\vert A\vert$ elements out of $B$, this can be done in $\binom{n}{m}$ different ways. Now put the chosen $m$ elements under the lined up elements of $A$. In how many different ways can you do this?

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    $\begingroup$ @J.G. Thank you for the edit! $\endgroup$ Commented Nov 5, 2019 at 7:47

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