June 9th, 2015, 08:40 AM  #1 
Member Joined: Apr 2015 From: Indonesia Posts: 53 Thanks: 2  Factorial Summation
Can someone tell me how to simplify the L.H.S form to R.H.S form?

June 9th, 2015, 08:57 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,690 Thanks: 2669 Math Focus: Mainly analysis and algebra 
I would guess that you should note that $${r! \over (ra)!} = a! {r \choose a}$$ and then use $${r \choose a} = {r 1 \choose a1} + {r 1 \choose a}$$ which is the algebraic way of writing the method you first used to construct Pascal's Triangle.

June 9th, 2015, 09:19 AM  #3 
Member Joined: Apr 2015 From: Indonesia Posts: 53 Thanks: 2  I do know that but I still can't simplify it to the R.H.S form. I have some trouble to simplify the factorial series.

June 9th, 2015, 10:18 AM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,690 Thanks: 2669 Math Focus: Mainly analysis and algebra 
Try rewriting the formula as $${r \choose a} = {r + 1 \choose a + 1}  {r \choose a + 1}$$ Have you guessed that you are looking for a telescoping sum? 
June 9th, 2015, 09:29 PM  #5 
Member Joined: Apr 2015 From: Indonesia Posts: 53 Thanks: 2  

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