I want to find how many surjective functions there are from the set $A=${$1,2,3,4,5$} to the set $B=${$1,2,3$}? I think the best option is to count all the functions ($3^5$) and then to subtract the non-surjective functions. However, I'm not sure how can I count these functions.
Thanks
I will show you two ways.
First one is with your current approach and using inclusion-exclusion, so you need to count the number of functions that miss at least $1$ element, lets call it $S_1$ which is equal to ${ 3 \choose 1 }2^5 = 96$. However, notice that this expression overcounts in the case of missing 2 elements. Everytime we count missing 1 and 3 for example, we also count missing 3 and 1 which are the same function. The number of functions that miss $2$ elements, call it $S_2$ is ${3 \choose 2}1^5 = 3$. Now, the total number of surjective functions is $3^5 - 96 + 3 = 150$.
But you can also do the following, fix a surjective function $f$ and consider the sets $f^{-1}(1), f^{-1}(2), f^{-1}(3)$. Because $f$ is surjective, they partition $A$ into $3$ disjoint, non empty sets.
Now think the other way around, start with $A$ and partition it into $3$ disjoint non empty sets, say $A_1, A_2, A_3$, you can then form a surjective function by just assigning one of the $A_i$ to one of the elements in $B$. The number of ways to partition a set of $n$ elements into $k$ disjoint nonempty sets are the Stirling numbers of the second kind, and the number of ways of of assigning the $A_i$ to the elements of $B$ is $k!$ (where $k$ is the size of $B$), in your particular case, this gives $3!S(5,3) = 150$.
The reason I showed you these two ways, is that you can use them to prove the "explicit" formula for the stirling numbers of the second kind, which is $$ k!S(n,k) = \sum_{j=0}^k (-1)^{k-j}{k \choose j} j^n $$ By just double counting, and using a more general inclusion exclusion, and as far as I know, this is one of the most "explicit" formulas you can get.
Pedestrian approach:
How many surjective functions from:
$A$ ={ $1, 2, 3, 4, 5$} to $B$= {$a, b, c$} ?
1) Let $3$ distinct elements of $A$ be mapped onto $a, b$, or $c$.
There are
$\binom{5}{3} = 10$,
ways to pick $3$ elements from $5$.
There are $3$ ways to map these elements onto $a,b$, or $c$.
Altogether $3×10 = 30$ ways.
Each choice leaves $2$ spots in $B$ empty; $2$ ways of filling the vacant spots with the $2$ remaining elements of $A$
Combining: $2×30 = 60$ ways of generating a surjectice map with $3$ elements mapped onto $1$ element of $B$.
2) $2$ elements of $A$ are mapped onto $1$ element of $B$, another $2$ elements of $A$ are mapped onto another element of $B$, and the remaining element of $A$ is mapped onto the remaining element of $B$.
The mapping looks, for example, like :
$( ||, |, || )$.
Let's start with the single element:
$5$ ways to choose an element from $A$, $3$ ways to map it to $a,b$ or $c$. Altogether: $5×3 =15$ ways.
$2$ vacant spots remain to be filled with $2$ elements of $A$ each.
$4$ elements are left in $A$, the number of ways of choosing $2$ of the remaining $4$: $ \binom{4}{2} = 6.$
To avoid double counting fix any one empty spot of $B$ (there are $2$).
There are $6$ ways to put $2$ numbers in this spot, the remaining open spot is taken care of with the remaining $2$ numbers of $A$ automatically.
Altogether there are $15×6 = 90$ ways of generating a surjective function that maps $2$ elements of $A$ onto $1$ element of $B$, another $2$ elements of $A$ onto another element of $B$, and the remaining element of $A$ onto the remaining element of $B$.
Combining: There are 60 + 90 = 150 ways.
