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If $a, b$ and $c$ are the greatest value of $^{19} \mathrm{C}_{\mathrm{p}},^{20} \mathrm{C}_{\mathrm{q}}$ and $^{21 }\mathrm{C}_{\mathrm{r}}$ respectively, then
$\frac{a}{11}=\frac{b}{22}=\frac{c}{21}$
$\frac{\mathrm{a}}{10}=\frac{\mathrm{b}}{11}=\frac{\mathrm{c}}{21}$
$\frac{\mathrm{a}}{10}=\frac{\mathrm{b}}{11}=\frac{\mathrm{c}}{42}$
$\frac{a}{11}=\frac{b}{22}=\frac{c}{42}$
Solution
$a=^{19} \mathrm{C}_{10}, \mathrm{b}=^{20} \mathrm{C}_{10}$ and $\mathrm{c}= ^{21} \mathrm{C}_{10}$
$\Rightarrow \mathrm{a}=^{19} \mathrm{C}_{9}, \mathrm{b}=2\left(^{19} \mathrm{C}_{9}\right)$ and $\mathrm{c}=\frac{21}{11}\left(^{20} \mathrm{C}_{10}\right)$
$\Rightarrow \mathrm{b}=2 \mathrm{a}$ and $\mathrm{c}=\frac{21}{11} \mathrm{b}=\frac{42 \mathrm{a}}{11}$
$\Rightarrow a: b: c=a: 2 a: \frac{42 a}{11}=11: 22: 42$
