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.






Using nCr notation,  where nCr = n!/(r!(n-r)!),  the problem is to show  

nCr + nC(r-1) = (n+1)Cr  (1)



Lets say you have n+1 elements:  {A1, A2, ... , An, An+1}



Put a star * on any one element.



The RHS of (1) is the selection of r elements from the set of n+1 elements.




One can choose these r elements in two mutually exclusive ways:



1. They can exclude the * element


This can be done in nCr ways  (all r elements are selected from the non-starred elements, and there are n of them).



2. They can include the * element


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)




Adding 1. and 2. gives  nCr + nC(r-1) = (n+1)Cr




For completeness, writing out the equations:

     







 

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