Talk:Factors of Binomial Coefficient/Corollary 2

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:

$\ds \paren {r - k} \binom r k = \paren {k+1} \binom r {k+1}$

but leave it. --Usagiop (talk) 19:31, 10 July 2023 (UTC)

All good thanks. I guess if current applications of the result are presented in the way seen here then it should stay this way, but if not then we could change it. Otherwise I would leave it be. Caliburn (talk) 19:42, 10 July 2023 (UTC)