document.write( "Question 1083531: Hello, I'm at a loss of how to solve this. Any help would be very much appreciated. Thanks in advance!\r
\n" ); document.write( "\n" ); document.write( "In Pascal's triangle, nCr+nCr+1=n+1Cr+1. Let X, Y, and Z be three consecutive numbers in the nth row of Pascal's Triangle, such that X = nCr, Y = nCr+1, and Z = nCr+2. Use the nature of Pascal's Triangle to show that X + Y + Z = n+2Cr+2 - nCr+1.\r
\n" ); document.write( "\n" ); document.write( "Hint: Start by finding X + Y and Y + Z.
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Algebra.Com's Answer #697545 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "To keep from getting bogged down with the n's and r's\r\n" );
document.write( "Let's write the \"Pascal's triangle formula\" using\r\n" );
document.write( "letters different from n and r:\r\n" );
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document.write( "sCt + sC(t+1) = (s+1)C(t+1)  <-- Pascal's triangle formula \r\n" );
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document.write( "X = nCr, Y = nCr+1, and Z = nCr+2\r\n" );
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document.write( "X + Y = nCr + nC(r+1) = (n+1)C(r+1)\r\n" );
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document.write( "Y + Z = nC(r+1) + nC(r+2) \r\n" );
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document.write( "Let s = n and t = r+1 in Pascal's triangle formula:\r\n" );
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document.write( "Y+Z = nC(r+1) + nC(r+1+1) = nC(r+1) + nC(r+2)\r\n" );
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document.write( "Adding the two equations together:\r\n" );
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document.write( "X + 2Y + Z = (n+1)C(r+1) + (n+1)C(r+2),\r\n" );
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document.write( "Now let s = n+1, and t = r+1 in Pascal's triangle formula:\r\n" );
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document.write( "(n+1)C(r+1) + (n+1)C(r+1+1) = (n+1+1)C(r+1+1) \r\n" );
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document.write( "which simplifies to\r\n" );
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document.write( "(n+1)C(r+1) + (n+1)C(r+2) = (n+2)C(r+2) \r\n" );
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document.write( "That left side turned out to be the same as the right side of \r\n" );
document.write( "X + 2Y + Z above. So\r\n" );
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document.write( "X + 2Y + Z = (n+1)C(r+1) + (n+1)C(r+2) = (n+2)C(r+2)\r\n" );
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document.write( "or\r\n" );
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document.write( "X + 2Y + Z = (n+2)C(r+2)\r\n" );
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\r\n" );
document.write( "To get X + Y + Z we need to subtract Y = nC(r+1) from\r\n" );
document.write( "both sides:\r\n" );
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document.write( "X + 2Y + Z = (n+2)C(r+2)\r\n" );
document.write( "     Y     =   nC(r+1)\r\n" );
document.write( "------------------------\r\n" );
document.write( "X  + Y + Z = (n+2)C(r+2) - nC(r+1)\r\n" );
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document.write( "Edwin
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