document.write( "Question 1189546: How many 9-digit codes can be formed if the code starts with 000 or ends with 1? \n" ); document.write( "

Algebra.Com's Answer #820932 by ikleyn(52790) $\"\"$ $\"About$

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\n" ); document.write( "How many 9-digit codes can be formed if the code starts with 000 or ends with 1?
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document.write( "To solve the problem, we should calculate separately\r\n" );
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document.write( "    - the number of the 9-digit codes that start with 000;\r\n" );
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document.write( "    - the number of the 9-digit codes that end with 1;\r\n" );
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document.write( "    - and then calculate the number of the codes in the union of these two sub-sets.\r\n" );
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document.write( "(1)  The number of the 9-digit codes that start with 000 is  = ,\r\n" );
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document.write( "     since the first three digits are just pre-determined, and only 6 remaining positions \r\n" );
document.write( "     each may have any of 10 digits.\r\n" );
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document.write( "(2)  The number of the 9-digit codes that end with  is  = ,\r\n" );
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document.write( "     since the last digit is just pre-determined, and only 8 remaining positions \r\n" );
document.write( "     each may have any of 10 digits.\r\n" );
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document.write( "(3)  To complete the solution, apply the formula for the union\r\n" );
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document.write( "          =  +  - .\r\n" );
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document.write( "       equals to  , since the intersection is the set of all codes\r\n" );
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document.write( "     that have pre-determined 000 in the first 3 positions and 1 in the last position,\r\n" );
document.write( "     having all 5 = 9-3-1 positions free for any of 10 digits.\r\n" );
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document.write( "     THEREFORE, the final formula and the final answer are\r\n" );
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document.write( "          =  +  -  = 1,000,000 + 100,000,000 - 100,000 = 100,900,000.\r\n" );
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document.write( "ANSWER.  There are  100,900,000  such codes.\r\n" );
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