Let $| X | = 7$ and $| Y | = 10$.
A- How many different one to one functions are there from $X$ to $Y$?
B- How many different functions are not one to one from $X$ to $Y$?
C- How many different one to one functions from $Y$ to $X$?
D- How many different functions are not one to one from $Y$ to $X$?
I have tired doing these but I think I'm doing them wrong I don't know I got. $4$ for $A$, $6$ for $B$, $0$ for $C$ and I don't know about the last one.
Just follow the definitions. $f: X \rightarrow Y$ $f$ 1-1. There are $10$ elements of $Y$ that the first element of $X$ can be mapped to. The second element must map to a different element so there are $9$ elements the second element can map to. And $8$ the third can map to. So there are $10*9*8*7*6*5*4$ ways to map the 7 elements of $X$ uniquely to the 10 elements of $Y$..
$f: X \rightarrow Y$ and $f$ not nescessarily 1-1. There are $10$ elements of $Y$ the first element of $X$ can be mapped to. The second element does not have to map to a different element so there are $10$ elements the second element can be mapped to. And $10$ the third can be mapped to. So there are $10^7$ ways to map the 7 elements of $X$ to the 10 elemments of $Y$. But if we don't want to count the 1-1 functions we must subtract $10*9*...*5*4$ that are one to one. So $10^7 - 10*9*....*54$ that are not 1-1.
I'll let you do the last two.