document.write( "Question 470975: Here is the question: What is the coefficient of w^2 x^3 yz^3 in the expansion of (w+x+y+z)^9? \n" ); document.write( "
Algebra.Com's Answer #323090 by Edwin McCravy(20060)\"\" \"About 
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Here is the question: What is the coefficient of w²x³yz³ in the expansion of (w+x+y+z)9\n" ); document.write( "
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document.write( "There are 9 4-term factors in (w+x+y+z)9\r\n" );
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document.write( "(w+x+y+z)(w+x+y+z)(w+x+y+z)(w+x+y+z)(w+x+y+z)(w+x+y+z)(w+x+y+z)(w+x+y+z)(w+x+y+z)\r\n" );
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document.write( "To multiply it all the way out we would choose 1 term from each\r\n" );
document.write( "factor of 4 terms.\r\n" );
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document.write( "To get w²x³yz³, \r\n" );
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document.write( "we choose the 2 w terms from the 9 4-term factors.\r\n" );
document.write( "That leaves 7 4-term factors from which to get the 3 x factors.\r\n" );
document.write( "We choose the 3 x terms from the remaining 7 4-term factors,\r\n" );
document.write( "That leaves 4 4-term factors from which to get the 1 y factor.\r\n" );
document.write( "We choose all 3 z terms from the remaining 3 4-term factors,\r\n" );
document.write( "and we're done.\r\n" );
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document.write( "So, going through it:\r\n" );
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document.write( "We can choose 2 of the 9 factors from which to get the two w's \r\n" );
document.write( "to make the w² in C(9,2) ways\r\n" );
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document.write( "For each of those C(9,2) ways, we can choose 3 of the remaining 7\r\n" );
document.write( "factors from which to get the 3 x's to make the x³ in C(7,3) ways.\r\n" );
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document.write( "That's C(9,2)C(7,3) ways to get w²x³\r\n" );
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document.write( "For each of those C(9,2)C(7,3) ways, we can choose 1 of the remaining 4\r\n" );
document.write( "factors from which to get the 1 y in C(4,1) ways.\r\n" );
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document.write( "That's C(9,2)C(7,3)C(4,1) ways to get w²x³y\r\n" );
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document.write( "For each of those C(9,2)C(7,3)C(4,1) ways, we can choose all 3 of the \r\n" );
document.write( "remaining 3 factors from which to get the 3 z's in C(3,3) ways, which\r\n" );
document.write( "is of course just 1 way.\r\n" );
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document.write( "That's C(9,2)C(7,3)C(4,1)C(3,3) ways to get a product of w²x³yz³\r\n" );
document.write( "choosing one letter from each of the 9 4-term factors.\r\n" );
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document.write( "So that's C(9,2)C(7,3)C(4,1)C(3,3) = 36*35*4*1 = 5040 ways.\r\n" );
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document.write( "So the coefficient of the term in w²x³yz³ is 5040.\r\n" );
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document.write( "Edwin
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