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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?
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June 9th, 2015, 08:57 AM   #2
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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.
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June 9th, 2015, 09:19 AM   #3
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Quote:
Originally Posted by v8archie View Post
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.
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June 9th, 2015, 10:18 AM   #4
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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?
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June 9th, 2015, 09:29 PM   #5
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Quote:
Originally Posted by v8archie View Post
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!
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