Suppose I have a trinomial distribution (or quadrinomial, or quintinomial, or... etc.) modeling all the outcomes that a three (or four, or five, or...) sided die can give in some number R of rounds.
Let's say that I want R=100 (that is, I roll a three sided die one hundred times) and I want to know what the probability that out of those hundred rolls, I get exactly 25 "one" results. I don't care about how many "two" or "three" results there are or what their proportion to each other is.
How do I do this? The Wikipedia page gives a very concise and clear explanation for how to find the probability of an outcome defined in terms of all the variables (in this case, that might be if I specified 25 "one"'s, 50 "two's", and 25 "three"'s), but what if I ONLY care about outcomes defined on one variable (or more generally, any number smaller than the total number of categories)?
This is simply a binomial random variable, where a success is "rolling a 'one'" and failure is "rolling something other than 'one'."