document.write( "Question 1135865: Show that n! /r!(n-r)! + n!/(r-1)! (n-r+1)! = (n+1)! /r!(n-r+1)! \n" ); document.write( "

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Show that n! /r!(n-r)! + n!/(r-1)! (n-r+1)! = (n+1)! /r!(n-r+1)!
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document.write( "Please be precise, looking at the first term,  n! / r!(n-r)!  is equal to (n!/r!)*(n-r)! which is probably NOT what you wanted.   You undoubtedly wanted n! / (r!(n-r)!).   Same for the other terms.  Be sure to use parentheses to convey the proper meaning.
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document.write( "Using nCr notation,  where nCr = n!/(r!(n-r)!),  the problem is to show  
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document.write( "nCr + nC(r-1) = (n+1)Cr  (1)\r
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document.write( "Lets say you have n+1 elements:  {A1, A2, ... , An, An+1}\r
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document.write( "Put a star * on any one element.\r
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document.write( "The RHS of (1) is the selection of r elements from the set of n+1 elements.
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document.write( "One can choose these r elements in two mutually exclusive ways:\r
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document.write( "1. They can exclude the * element

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document.write( "This can be done in nCr ways  (all r elements are selected from the non-starred elements, and there are n of them).\r
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document.write( "2. They can include the * element

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document.write( "This can be done in nC(r-1) ways (because the * element is pre-selected, this leaves a selection of the remaining r-1 elements from the remaining n elements)
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document.write( "Adding 1. and 2. gives  nCr + nC(r-1) = (n+1)Cr
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document.write( "For completeness, writing out the equations:
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