Suppose of all possible ways to choose two representatives among 10 students in a class, there are exactly 30 different ways to choose one or more female as a representative. How many males are there in the class?
+128407 We have this.....
C (F, 1) * C ( 10 - F , 1) + C (F,2) = 30 where F is the number of females
F! / ( F - 1)! * ( 10 - F)! / (9 - F)! + F! / [ (F-2)! * 2! ] = 30
F * ( 10 -F)! / ( 9 - F)! + (1/2) ( F)(F - 1) = 30
Note that F! / (F - 1)! = F and (10 - F)! / (9 - F)! = (10 - F)
So
F (10 - F) + (1/2)F (F-1) = 30
10F - F^2 + (1/2)F^2 - (1/2)F =30
(19/2)F - (1/2)F^2 = 30 multiply through by 2 and rearrange as
F^2 - 19F + 60 = 0 factor
(F - 4)( F -15) = 0
The first factor set to 0 provides the answer
F - 4 = 0
F = 4
