$\frac{1}{{9\choose r}} -\frac{1}{{10\choose r}} = \frac{11}{6{11\choose r}}$. Is there a way to find $r$ without using algebra?

Question

$$\frac{1}{\dbinom 9r} -\frac{1}{{\dbinom{10}r}} = \frac{11}{6\times \dbinom{11}r}$$

I guess directly applying algebra for this problem would be enough. But are there any simpler and prettier approaches to finding $r$ ?

Answer 1

Well, you can use Pascal's rule, or whatever it's called: $$ \binom{n}{k} + \binom{n}{k+1} = \binom{n+1}{k+1} $$ to rewrite the left-hand side as $$ \frac{\binom{10}{r}-\binom{9}{r}}{\binom{10}{r}\binom{9}{r}} = \frac{\binom{9}{r-1}}{\binom{10}{r}\binom{9}{r}}. $$ Multiplying up gives $$ \frac{11}{6}\binom{10}{r}\binom{9}{r} = \binom{9}{r-1}\binom{11}{r} $$ Our next trick is another well-known property of binomial coefficients, namely $$ \binom{n}{k+1}=\frac{n-k}{k+1}\binom{n}{k} $$ Use this to rewrite $$ \binom{9}{r} = \frac{10-r}{r}\binom{9}{r-1}, $$ so $$ \frac{11}{6}\binom{10}{r}\binom{9}{r} = 11\binom{10}{r}\frac{10-r}{r}\binom{9}{r-1} = \binom{9}{r-1}\binom{11}{r} \\ \frac{11}{6}\binom{10}{r}\frac{10-r}{r} = \binom{11}{r} $$ Now we have another binomial identity, $$ \binom{n+1}{k}=\frac{n+1}{n+1-k}\binom{n}{k}, $$ so $$ \binom{11}{6r} = \frac{11}{11-r}\binom{10}{r}, $$ reducing the equation to $$ \frac{1}{11-r} = \frac{10-r}{6r}, $$ which is the quadratic you get normally. This is really not inherently different, but does avoid getting factorials out.

Answer 2

This method is very simple :

$\frac{1}{{9\choose r}} -\frac{1}{{10\choose r}} =\frac{{10\choose r}-{9\choose r}}{{10\choose r}{9\choose r}}=\frac{\frac{{10\choose r}-{9\choose r}}{{9\choose r}}}{{10\choose r}}=\frac{(\frac{{10\choose r}-{9\choose r}}{{9\choose r}})\frac{11*6}{11-r}}{{10\choose r}\frac{11*6}{11-r}}=\frac{11}{6{11\choose r}}\Rightarrow \\ 11=(\frac{{10\choose r}-{9\choose r}}{{9\choose r}})\frac{11*6}{11-r}\Rightarrow \\ 1=(\frac{{10\choose r}-{9\choose r}}{{9\choose r}})\frac{6}{11-r}\Rightarrow \\ 1=(\frac{10}{10-r}-1)\frac{6}{11-r}\Rightarrow \\ 1=(\frac{r}{10-r})\frac{6}{11-r}\Rightarrow \\ r^2-21r+110=6r\Rightarrow...$