The number of way to distribute 30 identical candies among four children C₁, C₂, C₃ and C₄ so that C₂ receives atleast 4 and atmost 7 candies, C₃ receives atleast 2 and atmost 6 candies, is equal to :

$\begin{array}{l}Required number of ways=\\ \text{ Coefficient of }{\mathrm{x}}^{30}\text{ in }{\left(1+\mathrm{x}+{\mathrm{x}}^2+\dots +{\mathrm{x}}^{30}\right)}^2\\ \left({\mathrm{x}}^4+{\mathrm{x}}^5+{\mathrm{x}}^6+{\mathrm{x}}^7\right)\left({\mathrm{x}}^2+{\mathrm{x}}^3+{\mathrm{x}}^4+{\mathrm{x}}^5+{\mathrm{x}}^6\right)\\ {\mathrm{x}}^6{\left(\frac{1-{\mathrm{x}}^{31}}{1-\mathrm{x}}\right)}^2\left(1+\mathrm{x}+{\mathrm{x}}^2+{\mathrm{x}}^3\right)\left(1+\mathrm{x}+{\mathrm{x}}^2+{\mathrm{x}}^3+{\mathrm{x}}^4\right)\\ {\mathrm{x}}^6{\left(1-{\mathrm{x}}^{31}\right)}^2\left(1-{\mathrm{x}}^4\right)\left(1-{\mathrm{x}}^5\right)(1-\mathrm{x}{)}^{-4}\\ {\mathrm{x}}^6\left(1-{\mathrm{x}}^4-{\mathrm{x}}^5+{\mathrm{x}}^9\right)\left((1-2{\mathrm{x}}^{31}+.....)(1-\mathrm{x}{)}^{-4}\right)\\ {\mathrm{x}}^6\left(1-{\mathrm{x}}^4-{\mathrm{x}}^5+{\mathrm{x}}^9\right)(1-\mathrm{x}{)}^{-4}\\ \text{ Coefficient of }{\mathrm{x}}^{\mathrm{n}}\text{ in }(1-\mathrm{x}{)}^{-\mathrm{r}}{\text{ is }}^{\mathrm{n}+\mathrm{r}-1}{\mathrm{C}}_{\mathrm{r}-1}\\ {\Rightarrow }^{27}{\mathrm{C}}_3{-}^{23}{\mathrm{C}}_3{-}^{22}{\mathrm{C}}_3{+}^{18}{\mathrm{C}}_3\\ 2925-1771-1540+816\\ =430\end{array}$