C(n,j) = n!/((n-j)!j!)



Base case, n=1:



LHS:  C(j,j)+C(1+j,j) = 1 + (j+1)!/j! = j+2 

RHS:  C(1+j+1,j+1) = (j+2)!/(j+1)! = j+2



Base Case holds





Hypothesis:  Assume   

   C(j,j) + C(j+1,j) + C(j+2,j) + ... + C(n+j,j) = C(n+j+1, j+1)     (*)



is true for n=k.





Step case:  let n=k+1:  



...goal is to show LHS = RHS for n=k+1, making use of (*) when possible...



LHS:  + ... +  + C(k+j+1,j)



where the green terms represent the n=k case and can be replaced (using (*)) by C(k+j+1, j+1):



LHS =   C(k+j+1,j+1) + C(k+j+1,j)  

=  (k+j+1)!/(k!(j+1)!) + (k+j+1)!/((k+1)!j!)



...get this last expression over a common denominator... 



=  (k+1)(k+j+1)! / ((k+1)!(j+1)!) +  (k+j+1)!(j+1) / ((k+1)!(j+1)!)



...factor out (k+j+1)! from numerator...



=  (k+j+1)! ((k+1)+(j+1)) /  ((k+1)!(j+1)!)  



=   (k+j+2)! / ((k+1)!(j+1)!)         (1)



=   C(k+j+2, j+1)  



=  RHS



This shows (*) holds for n=k+1, and the proof is complete.

 



-------

Alternate way to show LHS = RHS...

 

RHS:  C((k+1)+j+1, j+1) = (k+j+2)!/((k+1)!(j+1)!)    ( = (1) )

 

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