If X is a set containing n elements and Y is a set containing m elements, how many functions are there from X to Y? How many of these functions are one-to-one?
For every element in X, I just need to associate an element of Y(for which I have m choices). Thus $\displaystyle m^n$ functions.
For one-one functions, For every element in X, I need to associate only one element of Y. Thus for the first one I have, m choices. Then for the second element, I have m-1 choices and so on. So it is $\displaystyle \frac{m!}{(m-n)!} $
Yes, it's the same.
But that's what I had in my notes. (they're from last year, so I don't have the proofs...)
I'm wondering... If $\displaystyle m\neq n$, is it possible to have one-to-one functions from X to Y ?
We may consider that we're dealing with functions whose domain is X, and whose range is Y. In which case, there is no one-to-one functions
If $\displaystyle m=n$, the number of bijective functions is n!