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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?

 Dec 28, 2020
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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

 

cool cool cool

 Dec 28, 2020
edited by CPhill  Dec 29, 2020

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