Proof for an alternative binomial coefficient

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December 22nd 2009, 08:45 AM
chris2547

Proof for an alternative binomial coefficient

Could anyone please point me in the direction on how to proof $C(-n, k) = (-1)^k C(n + k - 1, k)$ ?

I'm having a problem with this as the factorial of a negative integer is infinit.

Thanks in advance!
December 22nd 2009, 08:57 AM
novice

Quote:

Originally Posted by chris2547

Could anyone please point me in the direction on how to proof $C(-n, k) = (-1)^k C(n + k - 1, k)$ ?

I'm having a problem with this as the factorial of a negative integer is infinit.

Thanks in advance!

You mean $C(n,k)$ instead of $C(-n,k)$

Beside $(-1)^k$ is not infinity.
December 22nd 2009, 09:10 AM
chris2547

Quote:

Originally Posted by novice

You mean $C(n,k)$ instead of $C(-n,k)$

Beside $(-1)^k$ is not infinity.

No, I'm pretty sure I mean $C(-n,k)$ ; and did you ever try (-n!) ?
December 22nd 2009, 09:27 AM
CaptainBlack

Quote:

Originally Posted by chris2547

No, I'm pretty sure I mean $C(-n,k)$ ; and did you ever try (-n!) ?

I assume by $C(-n,k)$ you mean the coefficient of $x^k$ in the binomial expansion of $(1+x)^{-n}$

CB
December 22nd 2009, 09:56 AM
awkward

Quote:

Originally Posted by chris2547

Could anyone please point me in the direction on how to proof $C(-n, k) = (-1)^k C(n + k - 1, k)$ ?

I'm having a problem with this as the factorial of a negative integer is infinit.

Thanks in advance!

The starting point in this is to generalize the definition of C(x, k) to be

$C(x,k) = \frac{x (x-1) (x-2) \cdots (x-k+1)}{k!}$ .

With this definition, x can be any real number, not necessarily an integer. k must be a non-negative integer.

Now let x = -n and see what you get.
December 22nd 2009, 10:04 AM
chris2547

Quote:

Originally Posted by awkward

The starting point in this is to generalize the definition of C(x, k) to be

$C(x,k) = \frac{x (x-1) (x-2) \cdots (x-k+1)}{k!}$ .

With this definition, x can be any real number, not necessarily an integer. k must be a non-negative integer.

Now let x = -n and see what you get.

Thanks alot! I got it now. I just missed that one detail about the generalization of the binomial coëfficiënt.