Number of ways to watch one of 30 movies each day, and cover all genres I have tried solving this problem but my answer is not correct
Problem
In a cinema festival , 30 movies are scheduled every night in the following categories:


*

*10 in horror movies (H), 

*15 in comedy movies (C), 

*5  in documentaries (D)


A guest is allowed to watch 1 movie each night during the 5 night that he will be spending, he wish to watch a different movie every night, also he wish to watch a movie from each category during his stay in the festival.
In how many different way can he schedule his program.
My Answer 
First we make sure that he watch a movie in every category then we select 2 movies from the remaining 27 movies (all categories together) and since he can see those movies in different order during the 5 days we multiply everything by 5!
for ex:
({H1} , {C4}, {D3} , {H4,C13}) then we can permute this (5!) to get all arrangement possible
I would write that like this for all existing possibilities:
$$\binom{10}{1} \binom{15}{1}\binom{5}{1}\binom{27}{2} \cdot 5!$$
but my result is incorrect.
Can you please help, many thanks.
 A: Your result would be correct if, in addition to selecting which movies to watch, your guest were also identifying a favorite movie from each genre -- i.e., he would first pick a favorite from each of the three categories, and then pick two more movies from what's left over.  But since the problem doesn't ask for him to identify favorites, counting things this way leads to double (and even triple) counting.
I think you have to break things down into a bunch of cases:  The guest either picks three horror movies and one of each of the others, or three comedies and one of each of the others, or three documentaries and one of each of the others, or one horror movie and two of each of the others, or one comedy and ... I hope the rest is clear.  Can you take it from here?  (The $5$-factorial part of your solution was indeed correct, so don't forget to multiply by it at the end.)
A: Your method, of selecting one from each category then selecting any two others and permutating, will suffer from over counting.   Since it counts both all the ways to select, say "The Ring" as one of the ones, and all the ways to select it as one of the other two.

There are $5!$ ways to watch five different movies over the stay.   Now to choose which five, you must select at least one from each category.   That's either three from one and one from each other, or two from two and one from the other.   Sum all the ways to do this:
$$\newcommand{\Ch}[2]{{^{#1}\mathsf C_{#2}}}
5!\Biggl(\Ch{10}{3}\Ch{15}{1}\Ch{5}{1}+\Ch{10}{1}\Ch{15}{3}\Ch{5}{1}+\ldots\text{et cetera} \Biggr)
$$
