Combination/Permutation GRE question

[Idealistic]

There are 12 boys and 15 girls, each running for student council. There are two positions for the boys and two postions for the girls. If Karen, a girl, is guaranteed one postion on the student council, how many different ways can the student council positions be filled?

I figuered it would be 12*11 = 121 (since theres no repition) to fill the male positions

and 1*14 = 14 (since Karen is guaranteed a spot and it leaves 14 remaining girls) to fill the female positions

But I'm not sure what to do with these numbers; summing or multiplying them doesnt really mkaes sense.

 

[galactus]

Choose 2 of the boys from the 12. Then, there is only 1 girl to choose from the remaining 14 girls.

\(\displaystyle 14\binom{12}{2}\)

 

[JeffM]

There are 12 boys and 15 girls, each running for student council. There are two positions for the boys and two postions for the girls. If Karen, a girl, is guaranteed one postion on the student council, how many different ways can the student council positions be filled?

I figuered it would be 12*11 = 121 (since theres no repition) to fill the male positions Galactus of course gave you the correct answer, but it may be helpful to point out why your analysis is wrong. There are 12 ways to choose the first boy and 11 to choose the second, but Jim may be chosen first and Tom second or Tom first and Jim second. The order in which they are chosen is not relevant so multiplying 12 * 11 overstates the number of possible different combinations. Furthermore, 12 * 11 = 132, not 121.

and 1*14 = 14 (since Karen is guaranteed a spot and it leaves 14 remaining girls) to fill the female positions Correct

But I'm not sure what to do with these numbers; summing or multiplying them doesnt really mkaes sense.
What makes sense is to multiply the number of possible combinations of 1 Karen selected from the pool of 1 Karen (namely 1) times the number of possible combinations of 1 other girl selected from a pool of 14 female non-Karens (namely 14) times the number of possible combinations of 2 boys selected from a pool of 12 boys (namely 132/2 = 66).