How many ways can the director choose a quartet that contains at least one sopano if the chorus has six sopranos and eight baritones.?

There are 15 possibilities. In psuedo-code, the math works out like this: n = 42-1 where n is the number of possibilities, 4 is the size of a quartette, 2 is the number of different types of singers, and -1 reflects that a quartet of all non-sopranos does not meet your criteria.