Answer to Math Olympiad #1

Problem: 

I have four 3¢ stamps and three 5¢ stamps.  Using one or more of these stamps, how many different amounts of postage can I make?

Solution: 

The number of choices we have in using the 3¢ stamps is 5; we can use either 0, 1, 2, 3, or 4 of the 3¢ stamps.  Similarly, we have 4 choices with respect to the 5¢ stamps; we can use either 0, 1, 2, or 3 of the 5¢ stamps.  Each of the 5 choices for the 3¢ stamps can be combined with one of the four choices we have for the 5¢ stamps.  This gives a total of 20 combinations.  However, this total includes the combination of 0 3¢ stamps and 0 5¢ stamps.  Since 1 or more of the stamps must be used, we exclude the combination of none of each.  Therefore 19 different amounts of postage can be made.