Suppose I have a Multinomial random variable $X\sim Multinomial(n, p)$ i.e. $n$ trials, each trial having $P$ possible outcomes with values $a_1, a_2, ..., a_P$. Is there a nice expression for describing the probability that the sum of the random variable is greater than a given threshold $K$?
For concreteness, suppose I roll 100 6-sided dice. What's the probability that the sum of outcomes is greater than 500?