document.write( "Question 908843: Show that:
\n" ); document.write( "n C r = [(n-2) C r] + 2[(n-2) C (r-1)] + [(n-2) C (r-2)]\r
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Algebra.Com's Answer #551379 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"%22nCr%22\"\"%22%3F=%3F%22\"\"%22%28n-r%29Cr%22%2B2%2A%22%28n-2%29C%28r-1%29%22%2B%22%28n-2%29C%28r-2%29%22\"\r\n" );
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document.write( "We will work with the right side:\r\n" );
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document.write( "\"%28n-2%29%21%2F%28r%21%28n-2-r%29%21%29\"\"%22%22%2B%22%22\"\"2%2A%28n-2%29%21%2F%28r-1%29%21%28n%5E%22%22-2-%28r-1%29%29%21\"\"%22%22%2B%22%22\"\"%28n-2%29%21%2F%28r-2%29%21%28n%5E%22%22-2-%28r-2%29%29%21\"\r\n" );
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document.write( "\"%28n-2%29%21%2F%28r%21%28n-2-r%29%21%29\"\"%22%22%2B%22%22\"\"2%2A%28n-2%29%21%2F%28r-1%29%21%28n%5E%22%22-2-%28r-1%29%29%21\"\"%22%22%2B%22%22\"\"%28n-2%29%21%2F%28r-2%29%21%28n%5E%22%22-2-%28r-2%29%29%21\"\r\n" );
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document.write( "\"%28n-2%29%21%2F%28r%21%28n-2-r%29%21%29\"\"%22%22%2B%22%22\"\"2%2A%28n-2%29%21%2F%28r-1%29%21%28n-2-r%2B1%29%21\"\"%22%22%2B%22%22\"\"%28n-2%29%21%2F%28r-2%29%21%28n-2-r%2B2%29%21\"\r\n" );
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document.write( "Simplify and write letters before numbers:\r\n" );
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document.write( "\"%28n-2%29%21%2F%28r%21%28n-r-2%29%21%29\"\"%22%22%2B%22%22\"\"2%2A%28n-2%29%21%2F%28%28r-1%29%21%28n-r-1%29%21%29\"\"%22%22%2B%22%22\"\"%28n-2%29%21%2F%28%28r-2%29%21%28n-r%29%21%29\"\r\n" );
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document.write( "Factor out (n-2)!\r\n" );
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document.write( "Write r! as r(r-1)(r-2)!\r\n" );
document.write( "Write (r-1)! as (r-1)(r-2)!\r\n" );
document.write( "Write (n-r)! as (n-r)(n-r-1)(n-r-2)!\r\n" );
document.write( "Write (n-r-1)! as (n-r-1)(n-r-2)!\r\n" );
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document.write( "The LCD is \"r%28r-1%29%28r-2%29%21%28n-r%29%28n-r-1%29%28n-r-2%29%21\"\r\n" );
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document.write( "(1)  \"%28n-2%29%21%28%28+%28n-r%29%28n-r-1%29%2B2r%28n-r%29%2Br%28r-1%29%29%2F%28LCD%29%29+\"\r\n" );
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document.write( "Now we simplify the numerator of (1):\r\n" );
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document.write( "\"%28n-r%29%28n-r-1%29%2B2r%28n-r%29%2Br%28r-1%29\"\r\n" );
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document.write( "\"%28n-r%29%28n-r-1%29%2B2r%28n-r%29%2Br%28r-1%29\"\r\n" );
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document.write( "We factor (n-r) out of the first two terms:\r\n" );
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document.write( "\"%28n-r%29%28n-r-1%2B2r%29%2Br%28r-1%29\"\r\n" );
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document.write( "\"%28n-r%29%28n%2Br-1%29%2Br%28r-1%29\"\r\n" );
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document.write( "\"%28n-r%29%28%28n%2Br%29-1%5E%22%22%29%2Br%28r-1%29\"\r\n" );
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document.write( "Distribute the (n-r)\r\n" );
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document.write( "\"%28n-r%29%28n%2Br%29-%28n-r%29%2Br%28r-1%29\"\r\n" );
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document.write( "Multiply everything out to remove all the parentheses:\r\n" );
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document.write( "{{n^2-r^2-n+r+r^2-r}}}\r\n" );
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document.write( "The r's and rē's\r\n" );
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document.write( "\"n%5E2-n\"}\r\n" );
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document.write( "\"n%28n-1%29\"\r\n" );
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document.write( "Substitute for the numerator in (1)\r\n" );
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document.write( "(1)  \"%28n-2%29%21%28n%28n-1%29%29%2F%28LCD%29%29+\"\r\n" );
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document.write( "And when we multiply (n-2)! by that numerator,\r\n" );
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document.write( "(1)  \"%28n%28n-1%29%28n-2%29%21%29%2F%28LCD%29%29+\"\r\n" );
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document.write( "the numerator becones n!\r\n" );
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document.write( "(1)  \"n%21%2F%28LCD%29%29+\"\r\n" );
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document.write( "Next we simplify the LCD\r\n" );
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document.write( "LCD = \"r%28r-1%29%28r-2%29%21%28n-r%29%28n-r-1%29%28n-r-2%29%21\"\r\n" );
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document.write( "That is just \"r%21%28n-r%29%21\"\r\n" );
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document.write( "So now (1) becomes\r\n" );
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document.write( "(1)  \"n%21%2F%28r%21%28n-r%29%21%29\"\r\n" );
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document.write( "which is\r\n" );
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document.write( "(1)  \"nCr\"\r\n" );
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document.write( "Whew!\r\n" );
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document.write( "Edwin
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