A bag contains n identical red balls, 2n identical black balls and 3n identical white balls. If probability of drawing n balls of same colour is greater than or equal to1/6, then the minimum number of red balls in the bag is equal to ____

We have n identical red balls, 2n identical black
balls and 3 n identical white balls
Total number of selection of n balls from 2n and 3 n
balls, ${\mathrm{x}}_1+{\mathrm{x}}_2+{\mathrm{x}}_3=\mathrm{n}=$ Coefficient of
$\begin{array}{l}{\mathrm{x}}^{\mathrm{n}}\left(1+\mathrm{x}+{\mathrm{x}}^2+\dots .+{\mathrm{x}}^{\mathrm{n}}\right)\\ \left(1+\mathrm{x}+{\mathrm{x}}^2+\dots .+{\mathrm{x}}^{2\mathrm{n}}\right)\\ \left(1+\mathrm{x}+{\mathrm{x}}^2+\dots ..+{\mathrm{x}}^{3\mathrm{n}}\right)=\text{ Coefficient of }{\mathrm{x}}^{\mathrm{n}}\text{ in }\\ (1-\mathrm{x}{)}^{-3}=\mathrm{n}+3-1_{{\mathrm{C}}_{3-1}}=\mathrm{n}+2_{{\mathrm{C}}_2}\\ \text{ Favourable case }=1+1+1=3\\ \text{ Required probability }=3\left(\frac{2}{(\mathrm{n}+2)(\mathrm{n}+1)}\right)\ge \frac{1}{6}\\ \Rightarrow {\mathrm{n}}^2+3\mathrm{n}-34\ge 0\\ \mathrm{n}=\frac{-3\pm \sqrt{9+136}}{2}=\frac{-3\pm \sqrt{145}}{2}\\ =\frac{-3\pm 12.04}{2}>4.56\\ \mathrm{Min} \mathrm{value} \mathrm{of} \mathrm{n}=5\end{array}$