How many sets of $5$ episodes are there in which all four categories are purchased?

Question

There are $10$ drama episodes, $8$ comedy, $3$ news bulletins, and $6$ animations. A user is purchasing $5$ episodes. How many sets of $5$ episodes are there in which all four categories are purchased?

I have done the following :

$c(10,2)c(8,1)c(3,1)c(6,1) + c(10,1)c(8.2)c(3,1)c(6,1) + c(10,1)c(8,1)c(3,2)c(6,1) + c(10,1)c(8.1)c(3,1)c(6,2)$

which I calculate as a total of $16560$.

But thinking about it shouldn't it also be:

$c(10,1)c(8,1)c(3,1)c(6,1)c(23,1)$ which I calculate as $33120$.

So what am I doing wrong? Both answers make sense to me!

Answer 1

The first solution you wrote out is the correct one.

In the second solution, you are actually double-counting every possible outcome, as I'm sure you noticed that $33120$ was $2(16560)$.

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Let's see why this is happening. Using the process of the second solution, notice that we are choosing the last episode out of the remaining $23$. But if we fail to choose any particular episode the first time through (as in, choosing it before we choose the last episode from the $23$ remaining) we have the opportunity to pick it again as the fifth one. Just for a quick example, say we picked comedy episode $A$ in the first four, then comedy episode $B$ as the fifth one. Another way this method is counting is picking $B$ in the first four then $A$ as the fifth one, which happens to be the same combination of episodes. That means using this method we are double counting the number of outcomes.

Answer 2

Your first result is correct.

Your second method counts every selection twice, once for each way of designating one of the two episodes in the category from which two episodes are selected as the episode selected from that category and the other as the additional episode.

To make this concrete, suppose a person selects drama episodes $D_1$ and $D_2$, comedy episode $C_1$, news episode $N_1$, and animation episode $A_1$. You count this selection twice, once when you designate $D_1$ as the drama episode you are selecting and $D_2$ as the additional episode and once when you designate $D_2$ as the drama episode you are selecting and $D_1$ as the additional episode:

$$ \begin{array}{c c c c c} \text{drama} & \text{comedy} & \text{news} & \text{animation} & \text{additional}\\ D_1 & C_1 & N_1 & A_1 & D_2\\ D_2 & C_1 & N_1 & A_2 & D_1 \end{array} $$