# No. of One- One Functions that can be formed between two sets.

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)).

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.
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?