document.write( "Question 323371: If a number is selected at random from the set of all five-digit numbers in which the sum of the digits is equal to 43, compute the probability that this number will be divisible by 11.
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Algebra.Com's Answer #231451 by Edwin McCravy(20060) $\"\"$ $\"About$

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document.write( "Three of the five digits must be 9's because if we had fewer 9's, say even as\r\n" );
document.write( "many as two 9's, and the rest as large as they could possibly be, (that is, if\r\n" );
document.write( "they were all three 8's) you'd only have sum of digits 9+9+8+8+8=42 which is 1\r\n" );
document.write( "short of 43. So the three 9's each integer must have makes up 27 of the\r\n" );
document.write( "required total of 43, leaving 16 for the remaining two digits to have as a\r\n" );
document.write( "sum.  The only way for two digits to have sum 16 is 9+7 and 8+8. Therefore all\r\n" );
document.write( "possible such integers must be arrangements of the digits of either 79999 or of\r\n" );
document.write( "88999.\r\n" );
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document.write( "I.  Count the arrangements of the digits of the integer 79999.\r\n" );
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document.write( "There are 5 places to put the 7.  So there are 5 arrangements of the digits of\r\n" );
document.write( "79999.  Since there are so few we can list these.\r\n" );
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document.write( "1.  79999\r\n" );
document.write( "2.  97999 = 11*8909\r\n" );
document.write( "3.  99799 \r\n" );
document.write( "4.  99979 = 11*9089\r\n" );
document.write( "5.  99997\r\n" );
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document.write( "Only two of these five are divisible by 11, \r\n" );
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document.write( "II.  Count the arrangements of the digits of the integer 88999.\r\n" );
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document.write( "There are \"5 choose 2\" or 5C2 or  places to put the two\r\n" );
document.write( "8's.  So there are 10 arrangements of the \r\n" );
document.write( "digits of 88999.  That's not too many to list either.  Listing them:\r\n" );
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document.write( " 1. 88999\r\n" );
document.write( " 2. 89899\r\n" );
document.write( " 3. 89989\r\n" );
document.write( " 4. 89998\r\n" );
document.write( " 5. 98899\r\n" );
document.write( " 6. 98989 = 11*8999\r\n" );
document.write( " 7. 98998 \r\n" );
document.write( " 8. 99889\r\n" );
document.write( " 9. 99898\r\n" );
document.write( "10. 99988\r\n" );
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document.write( "We find that only one of these is divisible by 11.\r\n" );
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document.write( "So of the 5+10 or 15 positive integers with sum of digits 43,\r\n" );
document.write( "exactly 3 of them are divisible by 11.\r\n" );
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document.write( "So the desired probability is \"3 out of 15\" or  or  \r\n" );
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document.write( "Edwin

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