# wikipedia entry on Pascal's rule

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• October 5th 2011, 09:34 PM
issacnewton
wikipedia entry on Pascal's rule
Hi

here is link to Pascal's rule at wikipedia

Pascal's rule - Wikipedia, the free encyclopedia

its written as

$\binom{n}{k}+\binom{n}{k-1}=\binom{n+1}{k} \;\mbox{for}\;1\le k\le n+1$

if we plug in k=n+1 , we get

$\binom{n}{n+1}+\binom{n}{n}=\binom{n+1}{n+1}$

$\Rightarrow \binom{n}{n+1}=0$

is something wrong ?

(Emo)
• October 5th 2011, 09:39 PM
issacnewton
Re: wikipedia entry on Pascal's rule
i think the range should defined as

$1\le k\le n$

right ?
• October 5th 2011, 11:17 PM
CaptainBlack
Re: wikipedia entry on Pascal's rule
Quote:

Originally Posted by issacnewton
i think the range should defined as

$1\le k\le n$

right ?

How would you define

$\binom{n}{n+1}$?

It seems to involve (-1)! in the denominator which I'm sure Pascal would not approce of.

CB
• October 5th 2011, 11:21 PM
issacnewton
Re: wikipedia entry on Pascal's rule
thats what i was wondering.. is wikipedia entry wrong then ?
• October 6th 2011, 12:17 AM
CaptainBlack
Re: wikipedia entry on Pascal's rule
Quote:

Originally Posted by issacnewton
thats what i was wondering.. is wikipedia entry wrong then ?

I beleive so, >>Planet Math's<< limits for k are what we might expect

I think I should also give >>Proof Wiki's<< page a plug.

CB
• October 6th 2011, 03:14 AM
issacnewton
Re: wikipedia entry on Pascal's rule
Thanks captain , So did you fix 'proof wiki' page. When I went there, it was ok. Ask somebody at wikipedia to fix it.
• October 6th 2011, 04:34 AM
CaptainBlack
Re: wikipedia entry on Pascal's rule
Quote:

Originally Posted by issacnewton
Thanks captain , So did you fix 'proof wiki' page. When I went there, it was ok. Ask somebody at wikipedia to fix it.

Proof Wiki is nothing to do with Wikipedia, it is a site maintained by a sometime helper here on MHF.

CB
• October 8th 2011, 05:19 PM
awkward
Re: wikipedia entry on Pascal's rule
Some authors define
$\binom{n}{k}=0$
whenever it is not true that $0 \leq k < n$.

See, for example, Knuth, "The Art of Computer Programming, Vol 1", or
Binomial Coefficient -- from Wolfram MathWorld.

By this definition, it is correct to say that
$\binom{n}{n+1} = 0$.
• October 8th 2011, 05:42 PM
issacnewton
Re: wikipedia entry on Pascal's rule
if that is what the definition is , then all is well............
• October 8th 2011, 07:43 PM
CaptainBlack
Re: wikipedia entry on Pascal's rule
Quote:

Originally Posted by issacnewton
if that is what the definition is , then all is well............

It is not the definition but a definition. Knuth is particularly known for this sort of thing (defining undefined notation in a manner which suits his current purpose), but it is also consistent with interpretation of the factorial in terms of the gamma function.

CB