Fourteen graduate students have applied for four available teaching assistantships. In how many ways can these assistantships be awarded among the applicants if One particular student must be awarded an assistantship? The group of applicants includes eight men and six women and it is stipulated that at least one woman must be awarded an assistantship?

Question

anonymous · Answer

i am confused. is this one problem or two?

anonymous · Answer

Fourteen graduate students have applied for four available teaching assistantships. In how many ways can these assistantships be awarded among the applicants if One particular student must be awarded an assistantship? 

$\dbinom{13}{3}$ would seem to be the answer assuming of course that you cannot tell one assistantship from another

anonymous · Answer

One problem that has a two part answer. Solve for one, then solve for another.

anonymous · Answer

if this is a separate problem, 
The group of applicants includes eight men and six women and it is stipulated that at least one woman must be awarded an assistantship?
 
then the answer is 
$$\dbinom{6}{1}\dbinom{8}{3}+\dbinom{6}{2}\dbinom{8}{2}+\dbinom{6}{3}\dbinom{8}{1}+\dbinom{6}{4}$$

anonymous · Answer

or you can compute the total 
$$\dbinom{14}{4}$$ and subtract the number of ways to get no women 
$$\dbinom{8}{4}$$ so another way would be to write 
$$\dbinom{14}{4}-\dbinom{8}{4}$$

anonymous · Answer

Ok I just kept forgetting the (6 C 4) part. Thanks

anonymous · Answer

you can check that they are the same http://www.wolframalpha.com/input/?i=14+choose+4+-+8+choose+4 http://www.wolframalpha.com/input/?i=%286+choose+1%29*%28+8+choose+3%29+%2B+%286+choose+2%29*%28+8+choose+2%29%2B%286+c