document.write( "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? \n" ); document.write( "
Algebra.Com's Answer #334579 by Edwin McCravy(20055)\"\" \"About 
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document.write( "With the requirement of 1 40˘ stamp, that reduces the problem to\r\n" );
document.write( "\"How many different ways can she make 23˘ postage using a combination\r\n" );
document.write( " of 5˘, 3˘, and 2˘ stamps?\"\r\n" );
document.write( "For this problem, we need to know the rules of operations with even\r\n" );
document.write( "and odd numbers, sometimes known as the \"parity rules\":\r\n" );
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document.write( "EVEN+EVEN=EVEN       EVEN×EVEN=EVEN    \r\n" );
document.write( "EVEN+ODD=ODD         EVEN×ODD=EVEN\r\n" );
document.write( "ODD+ODD=EVEN         ODD×ODD=ODD \r\n" );
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document.write( "I. If she doesn't use any 5 cent stamps, then:\r\n" );
document.write( "      To make 23˘ with 3˘ and 2˘ stamps only, since 23 is odd, it can\r\n" );
document.write( "      only be written as an even number plus an odd number.\r\n" );
document.write( "      Any whole number of 2˘ stamps will produce an even number of\r\n" );
document.write( "      cents. Therefore to get an odd number to add to that even number to \r\n" );
document.write( "      make 23, she must use an odd number of 3˘ stamps. (That's because\r\n" );
document.write( "      an even number of 3˘ stamps would be an even number of cents.)\r\n" );
document.write( "      There are four odd numbers of 3˘ stamps possible (1,3,5, and 7).\r\n" );
document.write( "      That gives 4 ways since she can always make up the remaining even\r\n" );
document.write( "      part of 23˘ with 2˘ stamps.  So that's 4 ways.\r\n" );
document.write( "        \r\n" );
document.write( "2. If she uses exactly 1 5˘ stamp, then the problem reduces to\r\n" );
document.write( "   \"How many different ways can she make 18˘ postage using a combination\r\n" );
document.write( "   of 3˘ and 2˘ stamps?\"\r\n" );
document.write( "   \r\n" );
document.write( "      18 is even.  Any number of 2˘ stamps is even, so the 3˘ stamps\r\n" );
document.write( "      must also contribute an even number of cents, so she can only\r\n" );
document.write( "      use an even number of 3˘ stamps.  There are four even numbers \r\n" );
document.write( "      of 3˘ stamps possible (0,2,4, and 6). That gives 4 ways since she can\r\n" );
document.write( "      always make up the remaining even part of 23˘ with 2˘ stamps.  So \r\n" );
document.write( "      that's 4 more ways.  \r\n" );
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document.write( "3. If she uses exactly 2 5˘ stamp, then the problem reduces to\r\n" );
document.write( "   \"How many different ways can she make 13˘ postage using a combination\r\n" );
document.write( "   of 3˘ and 2˘ stamps?\"\r\n" );
document.write( "   \r\n" );
document.write( "      13 is odd.  Any number of 2˘ stamps is even, so the 3˘ stamps\r\n" );
document.write( "      must contribute an odd number of cents, so she can only\r\n" );
document.write( "      use an odd number of 3˘ stamps.  There are two odd numbers \r\n" );
document.write( "      of 3˘ stamps possible (1 and 3). That gives 2 ways since she can\r\n" );
document.write( "      always make up the remaining even part of 23˘ with 2˘ stamps.  So \r\n" );
document.write( "      that's 2 more ways.\r\n" );
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document.write( "4. If she uses exactly 3 5˘ stamp, then the problem reduces to\r\n" );
document.write( "   \"How many different ways can she make 8˘ postage using a combination\r\n" );
document.write( "   of 3˘ and 2˘ stamps<\"\r\n" );
document.write( "   \r\n" );
document.write( "      8 is even.  Any number of 2˘ stamps is even, so the 3˘ stamps\r\n" );
document.write( "      must contribute an even number of cents, so she can only\r\n" );
document.write( "      use an even number of 3˘ stamps. There are two even numbers \r\n" );
document.write( "      of 3˘ stamps possible (0 and 2). That gives us 2 ways since she can\r\n" );
document.write( "      always make up the remaining even part of 23˘ with 2˘ stamps.  So \r\n" );
document.write( "      that's 2 more ways.\r\n" );
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document.write( "5. If she uses exactly 3 5˘ stamp, then the problem reduces to\r\n" );
document.write( "   \"How many different ways can she make 3˘ postage using a combination\r\n" );
document.write( "   of 3˘ and 2˘ stamps?\"\r\n" );
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document.write( "      Obviously there is only 1 way -- 1 3˘ stamp and no 2˘ stamps. \r\n" );
document.write( "      That's 1 more way.\r\n" );
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document.write( "So the answer, adding the numbers of cases from above, is\r\n" );
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document.write( "4+4+2+2+1 = 13 ways.\r\n" );
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document.write( "Edwin
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