We have three different options here, and it's going to be easiest to just count them individually. We can have 3 romantic comedies and 2 horror movies, 4 romantic comedies and 1 horror movie, or all romantic comedies.

To start, let's count the ways we can choose 3 romantic comedies. There are 6 options for the first comedy, 5 for the second, and 4 for the third. That's a total of 6*5*4 = 120 different possibilities.

However, the order doesn't matter. Submitting movies A, B, and C is the same as submitting movies B, C, and A. So we need to divide out those repetitions. Each group of 3 can be arranged in 3! = 6 different ways, so we divide by 6 to get a total of \(\frac{120}{6} = 20\) different combinations of 3 romantic comedies to submit.

Now we have to pick 2 horror movies out of the list of 8. Following the same logic as before, we have 8 choices for the first movie, and 7 for the second, for a total of 8*7 = 56 possible pairs. However, each pair is counted twice. So we really only have 28 pairs.

Each of those 28 pairs of horror movies can be matched with any of the 20 groups of 3 comedies. So there are 28*20 = 560 options here in all.

Second, we could have 4 comedies and 1 horror movie. The calculation for the comedies will be \(\frac{6*5*4*3}{4*3*2*1} = 15\). There are 8 horror movies, which means we have 8 ways of picking 1 of them

Put those together for 15*8 = 120 different ways of choosing 4 comedies and 1 horror movie.

Next, if we choose all 5 romantic comedies, then we're only leaving one comedy behind. There are 6 ways to choose that one to be left behind, and so 6 ways of choosing 5 romantic comedies.

Add up all our possibilities, and we get 560 + 120 + 6 = 686 possibilities in all.