The chorus has six sopranos and eight baritones. in how many ways can the director choose a quartet that contains at least one soprano?

If you have six sopranos and eight baritones, you have fourteen musicians in total.
The number of possible quartets that you can get with 14 musician is:




Now, the number of possible quartets without a soprano that you can have is:




Therefore, the number of quarters that contains at list one soprano will be the number of possible quarters minus the number of quarters without a soprano:

We can conclude that the director can choose a quarter which contains at least one soprano in 931 ways. 

Dbc if u do the math and stuff that what i got let me know if i was wrong