# SOLUTION: Sheila has a lot of stamps in her drawer and will be mailing a number of 2-oz letters costing 66 of postage. How many different ways can she make 66 postage using a 40 stamp and co

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 Click here to see ALL problems on Mixture Word Problems Question 491204: Sheila has a lot of stamps in her drawer and will be mailing a number of 2-oz letters costing 66 of postage. How many different ways can she make 66 postage using a 40 stamp and combinations of 5˘, 3˘, and 2˘ stamps?Answer by Edwin McCravy(8999)   (Show Source): You can put this solution on YOUR website!```With the requirement of 1 40˘ stamp, that reduces the problem to "How many different ways can she make 23˘ postage using a combination of 5˘, 3˘, and 2˘ stamps?" For this problem, we need to know the rules of operations with even and odd numbers, sometimes known as the "parity rules": EVEN+EVEN=EVEN EVEN×EVEN=EVEN EVEN+ODD=ODD EVEN×ODD=EVEN ODD+ODD=EVEN ODD×ODD=ODD I. If she doesn't use any 5 cent stamps, then: To make 23˘ with 3˘ and 2˘ stamps only, since 23 is odd, it can only be written as an even number plus an odd number. Any whole number of 2˘ stamps will produce an even number of cents. Therefore to get an odd number to add to that even number to make 23, she must use an odd number of 3˘ stamps. (That's because an even number of 3˘ stamps would be an even number of cents.) There are four odd numbers of 3˘ stamps possible (1,3,5, and 7). That gives 4 ways since she can always make up the remaining even part of 23˘ with 2˘ stamps. So that's 4 ways. 2. If she uses exactly 1 5˘ stamp, then the problem reduces to "How many different ways can she make 18˘ postage using a combination of 3˘ and 2˘ stamps?" 18 is even. Any number of 2˘ stamps is even, so the 3˘ stamps must also contribute an even number of cents, so she can only use an even number of 3˘ stamps. There are four even numbers of 3˘ stamps possible (0,2,4, and 6). That gives 4 ways since she can always make up the remaining even part of 23˘ with 2˘ stamps. So that's 4 more ways. 3. If she uses exactly 2 5˘ stamp, then the problem reduces to "How many different ways can she make 13˘ postage using a combination of 3˘ and 2˘ stamps?" 13 is odd. Any number of 2˘ stamps is even, so the 3˘ stamps must contribute an odd number of cents, so she can only use an odd number of 3˘ stamps. There are two odd numbers of 3˘ stamps possible (1 and 3). That gives 2 ways since she can always make up the remaining even part of 23˘ with 2˘ stamps. So that's 2 more ways. 4. If she uses exactly 3 5˘ stamp, then the problem reduces to "How many different ways can she make 8˘ postage using a combination of 3˘ and 2˘ stamps<" 8 is even. Any number of 2˘ stamps is even, so the 3˘ stamps must contribute an even number of cents, so she can only use an even number of 3˘ stamps. There are two even numbers of 3˘ stamps possible (0 and 2). That gives us 2 ways since she can always make up the remaining even part of 23˘ with 2˘ stamps. So that's 2 more ways. 5. If she uses exactly 3 5˘ stamp, then the problem reduces to "How many different ways can she make 3˘ postage using a combination of 3˘ and 2˘ stamps?" Obviously there is only 1 way -- 1 3˘ stamp and no 2˘ stamps. That's 1 more way. So the answer, adding the numbers of cases from above, is 4+4+2+2+1 = 13 ways. Edwin```