document.write( "Question 1027444: All steps and explanations are appreciated! Thank you!!\r
\n" ); document.write( "\n" ); document.write( "In the set of three-digit integers {100,101,...,999} how many integers are there
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\n" ); document.write( "\n" ); document.write( "b) with three digits that are either non-decreasing (as in 477, 555, 123) or non-increasing (666,321,943)? (Note that digits may repeat.)
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Algebra.Com's Answer #642916 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "a1.  Digits in strictly ascending order:\r\n" );
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document.write( "It's 9 digits choose 3. 9C3 = 84.\r\n" );
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document.write( "That's because there is only one way to arrange\r\n" );
document.write( "3 digits in ascending order.\r\n" );
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document.write( "It's 9C3 not 10, because 0 cannot be used at all\r\n" );
document.write( "since the first digit must be at least 1, and\r\n" );
document.write( "the other two digits must be greater.\r\n" );
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document.write( "Answer: 84\r\n" );
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document.write( "a2. Digits in strictly descending order:\r\n" );
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document.write( "0 can be used, but it must come last. \r\n" );
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document.write( "Case 1.  No 0's\r\n" );
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document.write( "That's the same as a1, or 84, because we can\r\n" );
document.write( "rearrange each one in descending order.\r\n" );
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document.write( "Case 2.  There is a zero last.\r\n" );
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document.write( "This is the number of two digit numbers containing \r\n" );
document.write( "no zeros, because we can put a 0 on the right end \r\n" );
document.write( "of each and have a three-digit number in descending \r\n" );
document.write( "order.\r\n" );
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document.write( "The number of two digit numbers in descending\r\n" );
document.write( "order is 9 digits choose 2 or 9C2 = 36, because for\r\n" );
document.write( "every choice there is only one way to arrange them\r\n" );
document.write( "in descending order.  (Then we put a 0 at the end\r\n" );
document.write( "of each).\r\n" );
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document.write( "Answer: 84+36 = 120.\r\n" );
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document.write( "b1.  Digits in non-decreasing order:  \r\n" );
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document.write( "Note that we cannot use 0.\r\n" );
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document.write( "Case 1:  All three digits different.\r\n" );
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document.write( "It's the 84 in the first part above.\r\n" );
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document.write( "Case 2: Those with exactly two digits alike.\r\n" );
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document.write( "We can choose the digit to have two alike in 9 ways\r\n" );
document.write( "We can choose the digit that is not like the other\r\n" );
document.write( "two in 8 ways.  That's 9*8 = 72 ways.\r\n" );
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document.write( "Case 3: Those with all three digits the same.\r\n" );
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document.write( "They are 111,222,...,999\r\n" );
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document.write( "There are 9 of those.\r\n" );
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document.write( "Answer: 84+72+9 = 165 \r\n" );
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document.write( "b2.  Digits in non-increasing order.\r\n" );
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document.write( "Case 1: No zeros.\r\n" );
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document.write( "That the same as the 165 in b1 above, for we can arrange \r\n" );
document.write( "each one in non-increasing order. Notice that there were \r\n" );
document.write( "no 0's.\r\n" );
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document.write( "Case 2. Exactly 1 zero, which must be at the right end.\r\n" );
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document.write( "That's the same as the 36 of case 2 of a2, arranging the\r\n" );
document.write( "first two digits in non-increasing order, plus the 9 where\r\n" );
document.write( "the first two can be the same: 110,220,...,990\r\n" );
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document.write( "That's 36+9=45\r\n" );
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document.write( "Case 3. Exactly 2 zeros.\r\n" );
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document.write( "That's these 9:  100,200,...900\r\n" );
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document.write( "Answer: 165+45+9 = 219\r\n" );
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document.write( "Edwin
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