Talk:Factors of Binomial Coefficient/Corollary 2
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This is wrong I think, as a quick sanity check, $\ds \binom 5 {3 - 1}$ is 10 which is not divisible by $k = 3$. Specifically, the product should run up to $r - \paren {k - 1} + 1 = r - k + 2$ rather than $r - k$. So I think the correct result is $\ds \binom r {k - 1} = \frac k {r - k + 1} \binom r k$, which produces the correct $\ds \binom 5 2 = \binom 5 3$ and can be sanity checked for natural numbers easily enough. Caliburn (talk) 15:53, 10 July 2023 (UTC)
- I think, if $r\in\Z_{>0}$ and $k=r+1$, we need to assume $\frac 0 0 =1$ --Usagiop (talk) 19:18, 10 July 2023 (UTC)
Or you should write:
- $\ds \paren {r - k + 1} \binom r {k - 1} = k \binom r k$
--Usagiop (talk) 19:20, 10 July 2023 (UTC)
- Latter seems good, (I don't think we should make $0/0$ anything, only $0^0 = 1$ etc.) feel free to throw that in. Caliburn (talk) 19:21, 10 July 2023 (UTC)
- Fixed as above, but I would actually prefer:
- $\ds \paren {r - k} \binom r k = \paren {k+1} \binom r {k+1}$