probability of certain amount of things happening

i'm trying to figure out an easy way, if it exists, to calculate the probability of a certain amount of things (with different probabilities) happening. using college basketball as an example:

assume the probabilities of advancing to the ncaa tournament for each team are as follows. for simplicity's sake, we'll call these independent events:

ur .783

uri .804

temple .595

xavier .565

dayton .325

charlotte .097

gw .02

from this information, i can easily calculate that there is around a 7% chance of 5 teams advancing to the tournament, and less than a 1% chance of only 1 team advancing. i can also, of course, calculate that the expected number of teams advancing is 3.19.

what i'm having trouble with, however, is how to figure out how often exactly 2 teams, exactly 3 teams, and exactly 4 teams will advance. aside from manually multiplying out all of the possibilities and adding them together, which i'd rather avoid, is there an equation that accomplishes this?

thanks a lot for the help