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?

(a) No preference is given to any one student? 

A simple combination of four selections out of 14: 

14! / (10! 4!) = 14*13*12*11 / (4*3*2*1) 

= 7*13*11 = 1,001 

(b) One particular student must be awarded an assistantship? 

That leaves 3 selections out of 13 for the other positions: 

13! / (10! 3!) = 13*12*11 / (3*2*1) 

= 13*2*11 = 286 

(c) The group of applicants includes eight men and six women and it is stipulated that at least one woman must be awarded an assistantship? 

We have the original 1,001 possibilities from part (a), minus all of those that awarded all the positions to men: 

1,001 - 8! / (4! 4!) = 1,001 - 8*7*6*5 / (4*3*2*1) 

= 1,001 - 70 = 931

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