How many different ways can you distribute 5 apples and 8 oranges among six children? How many different ways can you distribute 5 apples and 8 oranges among six children if every child must receive at least one piece of fruit?
If there was a way to solve this using Pólya-Redfield that would be great, but I cannot figure out the group elements.

I am too lazy to calculate the numbers of elements with $k$ cycles in $S_8$ but if you do that yourself a solution could work as follows. (I will use this version of Redfield-Polya and $[n]$ shall denote $\{1,\dots,n\}$.)
Let us take $X = [13]$ the set of fruits, where $G= S_5 \times S_8$ acts on $X$ such that the first five apples and the later eight oranges are indistinguishable. Then
$$K_n = |[n]^X/G|$$
is the number of ways to distribute these apples and oranges among six distinguishable children. And
$$ N_n = K_n -n\cdot K_{n-1}$$
the of ways to distribute these apples and oranges among six distinguishable children such that every child must recieve at least one piece of fruit.
Now by the Theorem
$$K_n = \frac{1}{|G|} \sum_{g\in G} n^{c(g)} = \frac{1}{5!\cdot 8!} \left(\sum_{g\in S_5} n^{c(g)}\right)\left(\sum_{g\in S_8} n^{c(g)}\right) = \frac{1}{5!\cdot 8!} \left(\sum_{i\in [2]} d_i n^{i}\right) \left(\sum_{i\in [4]} e_i n^{i}\right),$$
where $c(g)$ is the number of cycles of $g$, $d_i$ the number of permutations of $S_5$ with exactly $i$ cycles and $e_i$ the number of permutations of $S_8$ with exactly $i$ cycles.
The number that we are looking for in the end is $N_6$.
