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

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

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

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

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

is something wrong ?

(Emo)

Re: wikipedia entry on Pascal's rule

i think the range should defined as

$\displaystyle 1\le k\le n$

right ?

Re: wikipedia entry on Pascal's rule

Quote:

Originally Posted by

**issacnewton** i think the range should defined as

$\displaystyle 1\le k\le n$

right ?

How would you define

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

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

CB

Re: wikipedia entry on Pascal's rule

thats what i was wondering.. is wikipedia entry wrong then ?

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

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.

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

Re: wikipedia entry on Pascal's rule

Some authors define

$\displaystyle \binom{n}{k}=0$

whenever it is not true that $\displaystyle 0 \leq k < n$.

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

$\displaystyle \binom{n}{n+1} = 0$.

Re: wikipedia entry on Pascal's rule

if that is what the definition is , then all is well............

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