document.write( "Question 588171: A four-digit number is to be formed using the digits 2,3,4,5,6,7, and 8 with no
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Algebra.Com's Answer #374400 by Edwin McCravy(20055)\"\" \"About 
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A four-digit number is to be formed using the digits 2,3,4,5,6,7, and 8 with no
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document.write( "The probability has a numerator and a denominator\r\n" );
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document.write( "We will calculate the numerator first, the number of successful cases and save\r\n" );
document.write( "the denominator till later (for \"dessert\") as it will be a cinch to calculate\r\n" );
document.write( "compared with the numerator.  \r\n" );
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document.write( "We break them into three categories:\r\n" );
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document.write( "1. The ones less than 7000 that end in 8.\r\n" );
document.write( "2. The ones less than 7000 that do not end in 8, (but instead end with 2, 4, \r\n" );
document.write( "   or 6).\r\n" );
document.write( "3. The ones more than 7000.\r\n" );
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document.write( "----------------------------\r\n" );
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document.write( "1.  To be even and not more than 7000, and end in 8, \r\n" );
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document.write( "We can choose the last (4th) digit just 1 way, as 8.\r\n" );
document.write( "We can then choose the 1st digit as any of the digits 2,3,4,5, or 6, \r\n" );
document.write( "which is 5 ways.\r\n" );
document.write( "We can then choose the 2nd digit as any of the 5 digits which we didn't \r\n" );
document.write( "choose for the 4th or 1st digit.\r\n" );
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document.write( "We can then choose the 3rd digit as any of the 4 digits we didn't \r\n" );
document.write( "choose for the 4th, 1st, or 2nd digits.\r\n" );
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document.write( "That's a total of 1·5·5·4 = 100 ways.\r\n" );
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document.write( "2. To be even and not more than 7000, and not end in 8, (ie. end in 2,4, or 6)\r\n" );
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document.write( "We can choose the 4th digit any of 3 ways.\r\n" );
document.write( "We can then choose the 1st digit as any of the 4 digits of {2,3,4,5,6} which\r\n" );
document.write( "we did not choose for the 4th digit.\r\n" );
document.write( "We can then choose the 2nd digit as any of the 5 digits we did not\r\n" );
document.write( "choose for the 4th or 1st digit.\r\n" );
document.write( "We can then choose the 3rd digit as any of the 4 digits we did not\r\n" );
document.write( "choose for the 4th, 1st, or 2nd digits.\r\n" );
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document.write( "That's 3·4·5·4 = 240 ways.\r\n" );
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document.write( "3. To be more than 7000, \r\n" );
document.write( "we can choose the 1st digit 2 ways 7, or 8\r\n" );
document.write( "We can then choose the 2nd digit as any of the remaining 6 digits.\r\n" );
document.write( "We can then choose the 3rd digit as any of the remaining 5 digits.\r\n" );
document.write( "We can then choose the 4th digit as any of the remaining 4 digits.\r\n" );
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document.write( "That's 2·6·5·4 = 240 ways.\r\n" );
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document.write( "Grand total = 100 + 240 + 240 = 580 ways.\r\n" );
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document.write( "So the numerator of the desired probability is 580\r\n" );
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document.write( "To calculate the denominator:\r\n" );
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document.write( "we can choose the 1st digit any of 7 ways\r\n" );
document.write( "We can then choose the 2nd digit as any of the remaining 6 digits.\r\n" );
document.write( "We can then choose the 3rd digit as any of the remaining 5 digits.\r\n" );
document.write( "We can then choose the 4th digit as any of the remaining 4 digits.\r\n" );
document.write( "That's 7·6·5·4 = 840\r\n" );
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document.write( "The desired probability is 580 ways out of 840 or \"580%2F840\"\r\n" );
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document.write( "That reduces to \"29%2F42\"\r\n" );
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document.write( "Edwin
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