Question:

Answer:
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a) 
The first award can be given to any of the 100 persons, once 
the first award is given then there are only 99 persons left
The second award can be given to any of the 99 persons, once 
the second award is given then there are only 98 persons left
The third award can be given to any of the 98 persons, once 
the third award is given then there are only 97 persons left
The fourth award can be given to any of the 97 persons, once 
the fourth award is given then there are only 96 persons left
so total no of ways = 100*99*98*97 = 94109400
b) 
if the grand prize is given to person who holds ticket 47
then there are three prizes which are to be distributed to 
3 people out of the 99 people left 
The second award can be given to any of the 99 persons, once 
the second award is given then there are only 98 persons left
The third award can be given to any of the 98 persons, once 
the third award is given then there are only 97 persons left
The fourth award can be given to any of the 97 persons, once 
the fourth award is given then there are only 96 persons left
so total no of ways = 99*98*97 = 941094
c)
if person holding ticket 47 gets one of the prizes then the rest 
3 prizes should be given to 3 out of 99 people 
The second award can be given to any of the 99 persons, once 
the second award is given then there are only 98 persons left
The third award can be given to any of the 98 persons, once 
the third award is given then there are only 97 persons left
The fourth award can be given to any of the 97 persons, once 
the fourth award is given then there are only 96 persons left
so total no of ways = 99*98*97 = 941094
d)
if person holding ticket 47 doesn't get any prize then 
the four prizes should be given to four people out of the 99 people left 
The first award can be given to any of the 99 persons, once 
the first award is given then there are only 98 persons left
The second award can be given to any of the 98 persons, once 
the second award is given then there are only 97 persons left
The third award can be given to any of the 97 persons, once 
the third award is given then there are only 96 persons left
The fourth award can be given to any of the 96 persons, once 
the fourth award is given then there are only 95 persons left
so total no of ways = 99*98*97*96 = 90345024