Jayne has 3 quarters, 2 dimes, 1 nickel and 2 pennies in her pocket. how many different amounts of money can she make using some or all of these coins?

Upper limit: There is a logical maximum that it could be. You can have 0, 1, 2, 3 quarters, 0, 1, 2 dimes, 0, 1 nickels, and 0, 1, 2 pennies. This is a total of 4*3*2*3 = 72. Of course, not all of these are distinct; this is why the answer is smaller. Also, this problem isn't counting $0.00 as a valid combination. Now that we have an idea of how the problems works, lets simplify it a little.

Pennies. Since you can have 0, 1, 2 pennies, there will be NO conflicts. (It takes 5 pennies to cause a problem, since then it will conflict with the nickel). It means that for all combinations of quarters, nickels, and dimes, we can have 0, 1, or 2 pennies. This means we can handle the other three coins and handle the pennies later. This brings the upper limit down to 24.

One combination has no nickels and no dimes, and another, equivalent combination has one fewer quarters, one nickel, and two dimes. There are three ways for the first combination: 1, 2, 3 quarters only. It will bring down the total to 21.

The pennies we have 21*3=63 possibilities.

General process:

Simplify the problem. (Pennies didn’t cause any conflict so they just make 3 times as many combination)

Determine the maximum possible combinations. This is a few multiplications. Not very difficult

Count the duplicates. This is often simpler because there are usually not that many of them. In this case, just three.

Put it back together.

Of course, sometimes there will be so many conflicts that anything short of listing them is unreasonable. This was quite a bit more difficult that it would have been to just list those 62 possibilities.