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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.
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.