Question

Find the number of non-negative integral solutions of $x_1+x_2+x_3+4x_4=20$.

Answer 1

The number of non-negative integral solutions of the given equation is equal to the coefficient of $x^{20}$ in $(1-x{)}^{-1}(1-x{)}^{-1}(1-x{)}^{-1}\times (1-x^4{)}^{-1}$, which is given by
Coefficient of $x^{20}$ in $(1-x{)}^{-3}(1-x^4{)}^{-1}$
$=$ Coefficient of $x^{20}$ in $(1{+}^3{C}_{1}x{+}^4{C}_2{x}^2{+}^5{C}_3{x}^3{+}^6{C}_4{x}^4+...)(1+x^4+x^8+...)$
$=1{+}^6{C}_4{+}^{10}{C}_8{+}^{14}{C}_{12}{+}^{18}{C}_{16}{+}^{22}{C}_{20}=536$