<s>[INST] How to find the value of $r$ such that $\frac{1}{\binom{9}{r}} - \frac{1}{\binom{10}{r}} = \frac{11}{6\binom{11}{r}}$? Find the value of $r$:
$$\frac{1}{\dbinom{9}{r}} - \frac{1}{\dbinom{10}{r}} = \frac{11}{6\dbinom{11}{r}}$$
I'm not sure where I should take this problem in order to isolate $r$. I seem to get many factorials that don't seem to combine. 
 [/INST] Making the problem more general
$$\frac{1}{\dbinom{n}{r}} - \frac{1}{\dbinom{n+1}{r}} = \frac{k}{\dbinom{n+2}{r}}$$ and using the same approach as N. F. Taussig in his/her answer, you should end with a quadratic equation in $r$
$$k r^2- \left(k(2  n+3 )+n+2\right)r+k\left( n^2+3  n+2\right)=0$$ If $k >0$, there are two positive roots $\left(\Delta=k^2+2 k (n+2) (2 n+3)+(n+2)^2,r_1+r_2>0,r_1r_2>0\right)$ and, as said in answers, you must discard the one which is > n.
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