Splitting up a million-dollar fortune into ten equal piles 
At his death, a millionaire left his 10 children a million dollars in cash, all in $\$100$, $\$10$, $\$1$ bills and $10$-cent and $1$-cent coins.  Show that there is a way for them to split the fortune into ten stacks of equal value. (Note that this would not be true if there were $3 bills.)

I'm struggling with how to get started. The assignment is on elementary set theory and functions, so there must be some way to apply these concepts, but I'm not sure how. The only thing that came to mind was inclusion-exclusion.
I would appreciate a hint on how to get started. I'll work on it more and edit this post with an updated attempt, but I'd really like to try to figure it out myself.
 A: The problem apperead as an optional review/warm-up assignment (not due) in MATH 55A.  The course provides a rigorous introduction to abstract algebra and I guess that the problem has been chosen as an unusual example of proof by induction.
We show a more general statement (for your case assume that 1 unit is equal to 1-cent coin).

Given an amount of money of $10^{n}$ in $10^{n-1},\dots, 1$ units
then there is a way to split it into $10$ stacks of equal value.

The proof is by induction on $n\geq 1$. The base step with $n=1$ is trivial: $1\cdot 10=10$ which means that each stack is of $1$ unit.
Inductive step: we assume that $n>1$ and
$$10^{n-1}b_{n-1}+\dots +10b_1+1b_0=10^n$$
where each $b_k$ is a non negative integer. By considering the equality modulo 10 we find that $b_0$ is a multiple of $10$. Then
$$10^{n-1}b_{n-1}+\dots +100b_2+10\left(b_1+\frac{b_0}{10}\right)=10^n$$
that is, after dividing both sides by 10,
$$10^{n-2}b_{n-1}+\dots +10b_2+\underbrace{\big(b_1+\frac{b_0}{10}\big)}_{\text{non-negative integer}}=10^{n-1}$$
and by the induction induction hypothesis we can divide $10^{n-1}$ into $10$ stacks of equal value.
A: I am not sure how much this approach uses set theory. Anyway this is the approach I found most natural. I give it as a hint.
Hint: order the money by value in decreasing order. First 100\$,10\$,1\$,0.1\$, and 0.01\$. Than start taking all banknotes with highest value to build the first heap. Show that you arrive exactly to 100 000\$ taking one by one the banknotes with highest value and therefore build the first heap. Then show that you can continue like that building all remaining nine heaps as well.
It is also instructive trying to generalize (e.g. does it work if we have just banknotes of 20 and 10 dollars?).
