1. (28 points) For the following question, please leave your answers in compact form. 100 students numbered 1, 2, 3, ..., 100, apply for a chance to win one of four distinct Oredigger scholarships or the prestigious Blaster scholarship, for a total of 5 scholarships. Winners are chosen at random. How many ways are there to award the scholarships if... (a) (4 points) there are no restrictions? Solution: (b) (4 points) applicant 63 wins the Blaster scholarship? Solution: (c) (4 points) applicants 8 and 25 both get scholarships? Solution: (d) (4 points) applicants 8, 25, 31, 78, and 97 all get scholarships? Solution: (e) (4 points) none of the applicants 8, 25, 31, 78, or 97 get a scholarship? Solution: (f) (4 points) the Blaster scholarship recipient is applicant 8, 25, 31, 78, or 97? Solution: (g) (4 points) applicants 31 and 78 get scholarships, but applicants 8, 25, and 97 do not get scholarships? Solution: