document.write( "Question 1122805: A seven-digit number (i.e. a whole number between 1000000 and 9999999) is selected at random. \r
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Algebra.Com's Answer #739288 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "First we count the number of 7 digit integers that contain no other\r\n" );
document.write( "digits than 6's, 7's, or 8's.\r\n" );
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document.write( "Choose the 1st digit 3 ways, either 6, 7, or 8.\r\n" );
document.write( "Choose the 2nd digit 3 ways, either 6, 7, or 8.\r\n" );
document.write( "Choose the 3rd digit 3 ways, either 6, 7, or 8.\r\n" );
document.write( "Choose the 4th digit 3 ways, either 6, 7, or 8.\r\n" );
document.write( "Choose the 5th digit 3 ways, either 6, 7, or 8.\r\n" );
document.write( "Choose the 6th digit 3 ways, either 6, 7, or 8.\r\n" );
document.write( "Choose the 7th digit 3 ways, either 6, 7, or 8. \r\n" );
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document.write( "That's 3∙3∙3∙3∙3∙3∙3 = 37 = 2187 ways\r\n" );
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document.write( "That's the numerator of the desired probability,\r\n" );
document.write( "the number of ways we can succeed in drawing a\r\n" );
document.write( "7 digit integers that contains no other digits \r\n" );
document.write( "than 6's, 7's, or 8's.\r\n" );
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document.write( "Now we count the number of ways to choose ANY\r\n" );
document.write( "7-digit number between 1000000 and 9999999:\r\n" );
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document.write( "Two ways to do that.  We could do it as above, but\r\n" );
document.write( "the easy way is to realize there are 9999999 integers'\r\n" );
document.write( "from 1 through 9999999, inclusive, and we eliminate\r\n" );
document.write( "those between 1 through 999999, those with fewer\r\n" );
document.write( "than 7 digits, and there are 999999 of those, so we\r\n" );
document.write( "subtract:\r\n" );
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document.write( "9999999\r\n" );
document.write( "-999999\r\n" );
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document.write( "9000000 = the denominator of the desired probability,\r\n" );
document.write( "the number of ways to succeed or fail.\r\n" );
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document.write( "Desired probability = 2187/9000000 which reduces by\r\n" );
document.write( "dividing top and bottom by 9:\r\n" );
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document.write( "Answer = 243/1000000\r\n" );
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document.write( "What is the probability it contains two 6's, three 7's, and two 8's?\r\n" );
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document.write( "There are 7 positions a digit in a 7-digit integer which a digit can\r\n" );
document.write( "occupy.\r\n" );
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document.write( "There are '7 positions choose 2' or 7C2=21 ways to fill in the two 6's.\r\n" );
document.write( "There then remain 5 positions to fill. \r\n" );
document.write( "There are '5 positions choose 3' or 5C3=10 ways to fill in the three 7's.\r\n" );
document.write( "There remain 2 positions to fill.\r\n" );
document.write( "There are '2 positions choose 2' or 2C2=1 way to fill in the two 8's.\r\n" );
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document.write( "Answer = (7C2)(5C3)(2C2) = 21∙10∙1 = 210 ways\r\n" );
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document.write( "Notice it would not have mattered if we had filled in the 7's first:\r\n" );
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document.write( "There are '7 positions choose 3' or 7C3=35 ways to fill in the three 7's.\r\n" );
document.write( "There then remain 4 positions to fill. \r\n" );
document.write( "There are '4 positions choose 2' or 4C2=6 ways to fill in the two 6's.\r\n" );
document.write( "There remain 2 positions to fill.\r\n" );
document.write( "There are '2 positions choose 2' or 2C2=1 way to fill in the two 8's.\r\n" );
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document.write( "Answer = (7C3)(4C2)(2C2) = 35∙6∙1 = 210 ways\r\n" );
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document.write( "That was not a necessary step, but just to show you that you can choose\r\n" );
document.write( "them in any order.\r\n" );
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document.write( "Desired probability = 210/9000000 which reduces by\r\n" );
document.write( "dividing top and bottom by 30:\r\n" );
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document.write( "Answer = 7/300000\r\n" );
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document.write( "Edwin
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