Question

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

Answer 1

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:
$14C4= \frac{14!}{4!(14-4)!}$
$14C4= \frac{14!}{(4!)(10!)}$
$14C4=1001$

Now, the number of possible quartets without a soprano that you can have is:
$8C4= \frac{8!}{4!(8-4)!}$
$8C4= \frac{8!}{(4!)(4!)}$
$8C4=70$

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:
$1001-70=931$
We can conclude that the director can choose a quarter which contains at least one soprano in 931 ways.