In simple probility theory one learns about tree diagrams and their rules. First you draw them with the outcomes. Then you write the probility of each event according to Laplace's formula and then to find the probility of one outcome you multiply every probability of each branch. If you are interested in multiple outcomes, you add the final probilities. So far so good.
This is intuitive and also simple if we have smaller trees. However with multiple outcomes, say already 20 outcomes which then are nested with other outcomes the process of (1) drawing (2) computing probability of each event (3) multiplying (4) adding, becomes very tedious very fast. Alone for drawing, the tree might take more than a page.
My question is now if there is an easier formula where you could just plug numbers in instead of always drawing a tree and writing every number. I just cannot imagine that mathematicians always have to draw a tree and do the other three steps every time they have to compute probilities of these kinds. (Obviously there are computers but I'm asking for a manual way.)
An analogy to make it clearer: Suppose you get exercises to compute the sum of numbers 1 to 90. Later you get another one to compute the sum from 1 to 104 and then the sum from 1 to 77. Now you would have to compute each same again and again like 1+2+3...+90 or 1+2+3...+104 which is very tedious. This is an analogy for the process of a tree diagram. However you could just use the formula n(n+1)/2, which holds for every case and you can easily reach the same result by such a simple computation formula without adding every integer. This small formula is an analogy for the "formula" or "process" I'm looking for instead of the tree digram.