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June 9th, 2015, 08:40 AM   #1
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Factorial Summation

Can someone tell me how to simplify the L.H.S form to R.H.S form?
Attached Images Screenshot_2.png (4.9 KB, 2 views) 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 $${r! \over (r-a)!} = a! {r \choose a}$$ and then use $${r \choose a} = {r -1 \choose a-1} + {r -1 \choose a}$$ which is the algebraic way of writing the method you first used to construct Pascal's Triangle. Thanks from topsquark and deadweight June 9th, 2015, 09:19 AM   #3
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Quote:
 Originally Posted by v8archie I would guess that you should note that $${r! \over (r-a)!} = a! {r \choose a}$$ and then use $${r \choose a} = {r -1 \choose a-1} + {r -1 \choose a}$$ which is the algebraic way of writing the method you first used to construct Pascal's Triangle.
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? Thanks from deadweight June 9th, 2015, 09:29 PM   #5
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Quote:
 Originally Posted by v8archie 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?
I used method of difference and it works! Thanks a lot, v8archie!  Tags factorial, summation factorialsummation

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