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I want to do 24C4 * 20C4 * 16C4* 12C4* 8C4* 4C4.

But I'm not sure if this keeps track of the flavors? All I know is that I'm calculating the possibilities of handing out 4 pieces of candy among 24 kids, then 4 among 20 kids...etc..

I'm not really looking at flavors? Or am I?

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    Try solving a smaller problem that you can check by listing every possible outcome. It's not only a way to check your method, but it can help to clarify your thoughts on a problem like this. – 2017-01-31
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    I think it is worth pointing out that your method works and does keep track of the flavours! If you want to think of it in a physical setting, what you are doing is taking the first flavour of candy, and then choosing $4$ of the $24$ students to take that flavour. Then, you are choosing the second candy, choosing $4$ of the $20$ students. Imagine being at the front of a class and starting with gum drops, then picking which 4 kids get those, then moving onto suckers, and choosing which 4 get those, etc. – 2017-01-31

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Should it be multinomial $\binom{24}{4,4,4,4,4,4}$? See this wiki on the interpretation.

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    We have not gotten that far in class yet :S. At least I don't think so. that might be the right way to interpret it, I just have never seen it done that way myself. – 2017-01-31
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Think of it this way.

Imagine the candy pieces in a row:

F1P1, F1P2, F1P3, F1P4, F2P1, ..., F6P4

where "F1P2" means "flavor 1 piece 2".

Then, place the kids in whatever order you want below that:

K5, K3, K21, K10, K6, ...

Then, you've given each kid a flavor (for example, flavor 1 went to kids 5, 3, 21, and 10).

There's 24! ways of ordering the kids, but they're redundant. There's 4! ways to rearrange the first four without changing the outcome, 4! ways to rearrange the second four, etc.

So your final answer is

$$\frac{24!}{4! 4! 4! 4! 4! 4!}$$

which by definition is the multinomial coefficient that Jan mentioned, $\binom{24}{4,4,4,4,4,4}$.

Your way of doing it does indeed come out to the same result, as mentioned in the linked Wikipedia page, and as you can see if you write out your version using the factorial definition and cancel.

Verifying that the result is the same in R:

> choose(24,4)*choose(20,4)*choose(16,4)*choose(12,4)*choose(8,4)*choose(4,4)
[1] 3.246671e+15
> factorial(24) / factorial(4)^6
[1] 3.246671e+15