NCr iff r>n

1. Feb 11, 2008

roger

what is nCr iff r>n?

what does it mean to choose say 5 objects from 4?

My understanding was that nCr is defined where n>r but I have a question involving this which I don't understand.

thanks
Roger

2. Feb 11, 2008

chroot

Staff Emeritus
By the defintion of the binomial coefficient "nCr," you'll get a number smaller than one when r > n. It's still "defined," but it doesn't make much sense to say you can choose 5 objects from 4 in 0.2 different ways.

- Warren

3. Feb 11, 2008

mathman

nCr=n!/(r!(n-r)!). If r>n, the denominator is oo, so nCr=0.

4. Feb 11, 2008

chroot

Staff Emeritus
Actually, mathman, you're right. I forgot that the subtraction on the bottom will result in a negative number... but the factorial of a negative number is undefined, and thus so is nCr where r > n.

- Warren

5. Feb 11, 2008

roger

what do you mean warren? mathman didnt say that he said If r>n, the denominator is oo, so nCr=0.

6. Feb 11, 2008

chroot

Staff Emeritus
If r > n, then the "(n-r)!" in the denominator is undefined. The factorial of negative numbers is undefined.

- Warren

7. Feb 11, 2008

roger

so why did he say it is equal to zero? and why did he say the denominator is oo?

8. Feb 11, 2008

chroot

Staff Emeritus
I don't know. Maybe you should ask him.

- Warren

9. Feb 11, 2008

roger

well it can't be undefined and zero at the same time it must be one or the other and if it is undefined I don't know why it is mentioned in the question I have.

10. Feb 11, 2008

John Creighto

11. Feb 11, 2008

chroot

Staff Emeritus
The gamma function is also undefined for negative real numbers. No matter how you slice it, "4 C 5" is undefined.

- Warren

12. Feb 11, 2008

John Creighto

Are you sure about that? The graph of the gamma function on wikipedia would suggest otherwise to me. Maybe I am misreading it.

13. Feb 11, 2008

d_leet

Negative inegers, I'm pretty sure the gamma function is defined for other negative numbers, just not the negative integers.

14. Feb 12, 2008

Mute

According to the wikipedia page on the binomial coefficient, the coefficient is defined to be zero for n < r (or r < 0 - there is a generalization to negative integers, though.).

http://en.wikipedia.org/wiki/Binomial_coefficient#Definition

Since this is just the definition of the binomial coefficient, not necessarily as it applies to combinatorics, one might argue that it doesn't apply to counting problems... However, it makes sense to me that there are zero ways to choose 5 objects from a set of 4, for instance.

Last edited: Feb 12, 2008
15. Feb 12, 2008

EnumaElish

Mathematica defines nCm as Binomial[n,m] = Γ(n+1)/(Γ(m+1)Γ(n-m+1)), seemingly = 0 when m > n.

If one defines nCm as combination with repetition, then the formula becomes (n+m-1)!/(m!(n-1)!), which does allow m > n.

16. Feb 12, 2008

mathman

(-1)!*0=0!=1, therefore (-1)!=1/0. 1/0 is not finite. For other negative integers, you will have a similar result.

17. Feb 12, 2008

chroot

Staff Emeritus
-1! is undefined, not infinite. That's what your equation indicates, too.

- Warren

18. Feb 12, 2008

roger

I'm getting varied answers and abit confused at that. What's the bottom line?

19. Feb 13, 2008

mathman

Look up Gamma function on Wikepedia (Gamma(n+1)=n!). You will see a graph as a function of real x. At 0 and each of the negative integers you will see a vertical line asymptote. On one side of each asymptote, the curve goes to +oo, on the other the curve goes to -oo. In that sense it is undefined (-oo or +oo), but certainly not finite. Therefore nCr for r>n is always 0.

20. Feb 13, 2008

EnumaElish

If r > n, nCr = 0 without repetition but nCr > 0 with repetition.