If $a, b$ and $c$ are the greatest values

We know ${ }^{n} C_{r}$ is greatest at middle term.

So, $a=\left({ }^{19} C_{p}\right)_{\max }={ }^{19} C_{10}={ }^{19} C_{9}$

$b=\left({ }^{20} C_{q}\right)_{\max }={ }^{20} C_{10}$

$c=\left({ }^{21} C_{6}\right)_{\max }={ }^{21} C_{10}={ }^{21} C_{11}$

Now, $\frac{a}{{ }^{19} C_{9}}=\frac{b}{\frac{20}{10} \cdot{ }^{19} C_{9}}=\frac{c}{\frac{21}{11} \cdot \frac{20}{10}{ }^{19} C_{9}}$

$\Rightarrow \quad \frac{a}{1}=\frac{b}{2}=\frac{c}{42 / 11}$ $\therefore \quad \frac{a}{11}=\frac{b}{22}=\frac{c}{42}$