document.write( "Question 672259: with the digits 1,2,3,4 and 5, how many five - digit positive inegers can be formed if the only the 3 can be repeated any number of times? \n" ); document.write( "
Algebra.Com's Answer #417878 by Edwin McCravy(20065)\"\" \"About 
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document.write( "If there is exactly 1 3,\r\n" );
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document.write( "We pick the 3's position any of C(5,1) ways, and\r\n" );
document.write( "We position the other 4 digits in P(4,4) ways. \r\n" );
document.write( "That's C(5,1)·P(4,4) ways.\r\n" );
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document.write( "If there are exactly 2 3's,\r\n" );
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document.write( "We pick the 3's position any of C(5,2) ways, and\r\n" );
document.write( "We position the other 3 digits left to right in P(4,3) ways.\r\n" );
document.write( "That's C(5,2)·P(4,3) ways.\r\n" );
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document.write( "If there are exactly 3 3's,\r\n" );
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document.write( "We pick the 3's position any of C(5,3) ways, and\r\n" );
document.write( "We position the other 2 digits left to right in P(4,2) ways.\r\n" );
document.write( "That's C(5,3)·P(4,2) ways.\r\n" );
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document.write( "If there are exactly 4 3's,\r\n" );
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document.write( "We pick the 3's position any of C(5,4) ways, and\r\n" );
document.write( "We position the other 1 digit in P(4,1) ways.\r\n" );
document.write( "That's C(5,4)·P(4,1) ways.\r\n" );
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document.write( "If there are exactly 5 3's, \r\n" );
document.write( "We pick the 3's position any of C(5,5) ways, and\r\n" );
document.write( "We position the other 0 digits in P(4,0) ways.\r\n" );
document.write( "That's C(5,5)·P(4,0) ways.\r\n" );
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document.write( "[Incidentally if there are exactly 5 3's, that works out to\r\n" );
document.write( "be only one way, 33333.]\r\n" );
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document.write( "Answer: \r\n" );
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document.write( "C(5,1)·P(4,4) + C(5,2)·P(4,3) + C(5,3)·P(4,2) + C(5,4)·P(4,1) + C(5,5)·P(4,0) =\r\n" );
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document.write( "5·24 + 10·24 + 10·12 + 5·4 + 1·1 = 120 + 240 + 120 + 20 + 1 = 501 ways\r\n" );
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document.write( "Edwin
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