Counting total and onto functions The question is :
How many total and onto functions there are from 
$X = \{1,2,3,4,5,6,7,8\}$ to $Y = \{a,b,c,d\}$.
I'm pretty sure you need to use the Inclusion - Exclusion method here but not really sure how.
Thanks in advance !
 A: A total function maps each element of $X$ to an element of $Y$.  Since there are four elements in $Y$, there are four ways we can map each of the eight elements in $X$.  Thus, there are $4^8$ total functions from $X$ to $Y$.
We do need to use the Inclusion-Exclusion Principle for the second question since we must exclude those functions that map to fewer than four elements of $Y$.  There are $4^8$ functions from $X$ to $Y$.  The number of functions that map to three elements of $Y$ is 
$$\binom{4}{3}3^8$$
since there are $\binom{4}{3}$ ways of selecting which of the three elements of $Y$ are in the range and three ways we can map each of the eight elements in $X$.  By similar reasoning, there are 
$$\binom{4}{2}2^8$$
functions whose range is two of the four elements in $Y$ and 
$$\binom{4}{1}1^8$$
functions whose range is one of the four elements in $Y$.  Thus, by the Inclusion-Exclusion Principle, the number of surjective (onto) functions from $X$ to $Y$ is 
$$4^8 - \binom{4}{3}3^8 + \binom{4}{2}2^8 - \binom{4}{1}1^8$$
