document.write( "Question 947191: Show that the way in which the entries in Pascal's triangle are formedby adding \"above left\" and \"above right' is consistent with the following statement;
\n" ); document.write( "nCr-1 +nCr =n+1Cr\r
\n" ); document.write( "\n" ); document.write( "Use the fact that nCr = n!/(n-r)!r! to show that the above statement is true. \n" ); document.write( "
Algebra.Com's Answer #578039 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
Question 940228
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document.write( "nC(r-1) + nC(r) ?=? (n+1)Cr\r\n" );
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document.write( "(1)   nCr = \"n%21%2F%28r%21%28n-r%29%21%29\"\r\n" );
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document.write( "(2)   nC(r-1) = \"n%21%2F%28%28r-1%29%21%28n%5E%22%22-%28r-1%29%29%21%29%29\" = \"n%21%2F%28%28r-1%29%21%28n-r%2B1%29%21%29\"\r\n" );
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document.write( "(3)   (n+1)Cr = \"%28n%2B1%29%21%2F%28r%21%28%28n%2B1%29%5E%22%22-r%29%21%29%29\" = \"%28n%2B1%29%21%2F%28r%21%28n%2B1-r%29%21%29\"\r\n" );
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document.write( "We want to prove that expression(2) + expression(1) = expression(3)\r\n" );
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document.write( "nC(r-1) + nCr =\r\n" );
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document.write( "\"n%21%2F%28%28r-1%29%21%28n-r%2B1%29%21%29\"\"%22%22%2B%22%22\"\"n%21%2F%28r%21%28n-r%29%21%29\"\r\n" );
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document.write( "Substitute (n-r+1)(n-r)! for (n-r+1)! and r(r-1)! for r!\r\n" );
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document.write( "\"n%21%2F%28%28r-1%29%21%28n-r%2B1%29%5E%22%22%28n-r%29%21%29\"\"%22%22%2B%22%22\"\"n%21%2F%28r%28r-1%29%21%28n-r%29%21%29\"\r\n" );
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document.write( "LCD = r(r-1)!(n-r+1)(n-r)!\r\n" );
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document.write( "\"n%21%2Ar%2F%28%28r-1%29%21%28n-r%2B1%29%28n-r%29%21r%29\"\"%22%22%2B%22%22\"\"n%21%28n-r%2B1%29%2F%28r%28r-1%29%21%28n-r%29%21%28n-r%2B1%29%21%29\"\r\n" );
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document.write( "replace r(r-1)! by r! and replace (n-r+1)(n-r) by (n-r+1)!\r\n" );
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document.write( "\"+n%21r%2F%28+r%21%28n-r%2B1%29%21%29++\"\"%22%22%2B%22%22\"\"n%21%28n-r%2B1%29%2F%28r%21%28n-r%2B1%29%21%29\"\r\n" );
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document.write( "\"%28n%21r%2Bn%21%28n-r%2B1%29%29%2F%28r%21%28n-r%2B1%29%21%29\"\r\n" );
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document.write( "\"%28n%21r%2Bn%21n-n%21r%2Bn%21%29%2F%28r%21%28n-r%2B1%29%21%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"%28cross%28n%21r%29%2Bn%21n-cross%28n%21r%29%2Bn%21%29%2F%28r%21%28n-r%2B1%29%21%29\"\r\n" );
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document.write( "\"%28n%21n%2Bn%21%29%2F%28r%21%28n-r%2B1%29%21%29\"\r\n" );
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document.write( "\"%28n%2B1%29n%21%2F%28r%21%28n-r%2B1%29%21%29\"\r\n" );
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document.write( "\"%28n%2B1%29%21%2F%28r%21%28n-r%2B1%29%21%29\"\r\n" );
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document.write( "(n+1)Cr\r\n" );
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document.write( "Edwin
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