Nickels and Trig IDs

I'm bored and was trying to explain Trig IDs to a student. One analogy I came up with was trying to make 25 cents from pennies and nickels. If you know that a nickel is five cents, and a penny is one, you can make 25 cents.

And then I got bored so I made this:

Twenty-five cents:
penny, nickel, dime, quarter.
= 25p
= 20p + 1n
= 15p + 2n
= 10p + 3n
= 5p + 4n
= 5n
= 3n + 1d
= 1n + 2d
= 1q


Recently, my brother had asked me what is the process for writing out all the combinations of elements in sets? For example...

Set A consists of these elements:
a₁, a₂, a₃.

Set B:
b₁, b_2.

Set C:
c₁, c₂, c₃, c₄.

and I want to write out every combination possible; so:

a₁, b₁, c₁;
a₁, b₁, c₂;
a₁, b₁, c₃;
a₁, b₁, c₄;
a₁, b₂, c₁;
a₁, b₂, c₂;
a₁, b₂, c₃;
a₁, b₂, c₄;
a₂, b₁, c₁;
a₂, b₁, c₂;
a₂, b₁, c₃;
a₂, b₁, c₄;
a₂, b₂, c₁;
a₂, b₂, c₂;
a₂, b₂, c₃;
a₂, b₂, c₄;
a₃, b₁, c₁;
a₃, b₁, c₂;
a₃, b₁, c₃;
a₃, b₁, c₄;
a₃, b₂, c₁;
a₃, b₂, c₂;
a₃, b₂, c₃;
a₃, b₂, c₄.

What is the process my brain goes through?

My brother wanted to know the process, because he was writing a computer program, and wanted to make it shorter--fewer lines of code. I'm not a programmer, so I don't know/remember exactly what his code says, but it does something like this:

For every a₁,
for every b₁,
write out an element of Set C.

For every a₁,
for every b₂,
write out an element of Set C.

...

Something terrible like that. And he doesn't just have A, B and C; he has about fourteen of these, so his code gets pretty dang long!

I thought; and thought; and thought about this, but I couldn't simplify it.

And then, I realized, no matter how many (few) elements in each set, there will be more rows than columns (assuming it's organized this way). So, instead of filling out the "chart" one row at a time, maybe it would be shorter to fill it out one column at a time! You'd just have to tell your program how many times to write each element before moving to the next element in the set (eg: how many times to write "a₁" before writing "a₂", etc), and when to stop, I guess, if programs require that.


But what is the method to write all the combinations of coins needed to make some amount of money? Wooo, Canadian monies.


Man am I tired,
--Charissa