a) How many different sums of money can be made from a toonie, a $5 bill and a $10 bill.

I think this equals 2^3-1

b) How many different sums of money can be made from the money in (a) as well as one more $10 bill?

This should equal 11.

c) Why does the situation become much more complicated if another $5 bill is added?

To answer these questions, we can use the concept of combinations. Let's break it down step by step:

a) In this case, we have three different bills: a toonie, a $5 bill, and a $10 bill. To find the number of different sums of money that can be made using these bills, we can consider it as a combination problem.

The number of ways to choose or not choose each individual bill is 2 (we can either include it or not include it). Since we have three bills, the total number of combinations is 2^3 = 8.

However, we need to subtract 1 from this number because we cannot have a sum of zero. So the answer is 8 - 1 = 7.

b) Now let's consider the situation where we add one more $10 bill. Similar to the previous question, we have to consider the combinations of choosing or not choosing each bill (including the additional $10 bill).

With four bills to consider (toonie, $5 bill, $10 bill, and additional $10 bill), the number of combinations is 2^4 = 16. Again, we need to subtract 1 because we cannot have a sum of zero. So the answer is 16 - 1 = 15.

c) Adding another $5 bill complicates the situation because it introduces more possibilities into the combinations. Now we have five bills: toonie, two $5 bills, $10 bill, and additional $10 bill.

Considering all possible combinations of choosing or not choosing each bill, the total number of combinations will increase significantly. To calculate the exact number, we need to go through all the possible combinations and calculate them manually.

Alternatively, we can use a more advanced concept called partitions, which calculates the number of possible combinations for a given set of values. However, this can be quite complex to understand and calculate without specialized tools or algorithms.

So, in short, the situation becomes more complicated when an additional $5 bill is added because it significantly increases the number of possible combinations, which becomes challenging to determine without an automated tool or algorithm.