I suppose this is a computer science course you are taking, for there
is no way this can be solved with ordinary counting methods of mathematics
courses.  I used LibertyBasic with the program below.  You can download a 
free trial copy of LibertyBasic at

http://www.libertybasic.com/ 

The output was 4563

---------------------------------------

for pound = 0 to 1
for fifty = 0 to (100-100*pound)/50
for twenty = 0 to (100-100*pound-50*fifty)/20
for ten = 0 to (100-100*pound-50*fifty-20*twenty)/10
for five = 0 to (100-100*pound-50*fifty-20*twenty-10*ten)/5
for two = 0 to (100-100*pound-50*fifty-20*twenty-10*ten-5*five)/2
 DISABLED_event_one=  100-100*pound-50*fifty-20*twenty-10*ten-5*five-2*two
count=count+1
next
next
next
next
next
next
print count

Edwin