Multinomial problem

I have five possible values (1-5) that are to be applied to categories regarding the likelihood, severity, etc, of systems problem events. There are twelve risk aspects that use the 1-5 rating.

I have determined the probabilities of individual category ratings to be:

1 - 7%
2 - 18%
3 - 45%
4 - 20%
5 - 10%

I need to find some way of determining an overall rating of 1-5, assuming a random distribution based on the above probabilities. In other words, a particular risk, e.g. power outage, is rated 1-5 in those 12 categories. I could just sum up the ratings numbers, I suppose, but how *should* I determine the overall rating?

It seems I should do a multinomial probability distribution to find the probability that certain numbers would occur, look at the cumulative and then make a subjective judgment on what I am going to call a 1, a 2, etc. But... I don't know how to do the multinomial calculations. I can do binomial, and I have tables and excel formulae for that... and I know this is an extension of that. I've even done the "fit" test. But, what I really want is to build a distribution of the possible summed values (min 12, max 60 if I sum them) based on the probabilities.

Any help is GREATLY appreciated!!