document.write( "Question 1143956: Proof that (n+1combination r) is equal to (n combination r-1) + (n combination r) \n" ); document.write( "
Algebra.Com's Answer #764912 by Edwin McCravy(20065)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "C(n+1,r) = C(n,r-1) + C(n,r)\r\n" );
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document.write( "which is often written this way:\r\n" );
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document.write( "We'll use the facts that \r\n" );
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document.write( "\"%28matrix%282%2C1%2CM%2CS%29%29=+M%21%2F%28S%21%28M-S%29%21%29\", \"M%21=M%2A%28M-1%29%21\" and \"%28M%2B1%29%21=%28M%2B1%29M%21\"\r\n" );
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document.write( "We'll prove the right side equals the left side:\r\n" );
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document.write( "\"%28matrix%282%2C1%2Cn%2Cr-1%29%29%2B%28matrix%282%2C1%2Cn%2Cr%29%29\"\r\n" );
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document.write( "\"matrix%283%2C1%2C%22%22%2C%22%22=%22%22%2C%22%22%29\"\r\n" );
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document.write( "Rewrite (n-r+1)! in the left denominator and r! in the right denominator:\r\n" );
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document.write( "The LCD is r(r-1)!(n-r+1)(n-r)!\r\n" );
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document.write( "Factor n! out of the numerator and rewrite the denominator:\r\n" );
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document.write( "\"%28n%21%28r%5E%22%22%2B%28n-r%2B1%29%29%29%2F%28r%21%5E%22%22%28n-r%2B1%29%21%29\"\r\n" );
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document.write( "\"%28n%21%28r%5E%22%22%2Bn-r%2B1%29%29%2F%28r%21%5E%22%22%28n-r%2B1%29%21%29\"\r\n" );
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document.write( "\"%28n%21%5E%22%22%28n%2B1%29%29%2F%28r%21%5E%22%22%28n-r%2B1%29%21%29\"\r\n" );
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document.write( "Rewrite the numerator as (n+1)! and the (n-r+1)! as (n+1-r)!\r\n" );
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document.write( "\"%28n%2B1%29%21%2F%28r%21%5E%22%22%28n%2B1-r%29%21%29\"\r\n" );
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document.write( "That's the same as the original left side:\r\n" );
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document.write( "\"%28matrix%282%2C1%2Cn%2B1%2Cr%29%29\"\r\n" );
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document.write( "Edwin
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