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= (14!)/(4!(14-4)!)`

`14C4= (14!)/((4!)(10!))`

`14C4=1001`

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

`8C4= (8!)/(4!(8-4)!)`

`8C4= (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.