Question

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

Answer 1

Number of non-negative integral solutions of the given equation
=coefficient of $x^{20}$ in ${(1-x)}^{-1}{(1-x)}^{-1}{(1-x)}^{-1}{(1-x^4)}^{-1}$
=coefficient of $x^{20}$ in ${(1-x)}^3{(1-x^4)}^{-1}$
=coefficient of $x^{20}$ in
$(1{+}^3{C}_{1}x{+}^4{C_2{x}^2+}^5{C_3{x}^2+}^6{C_4{x}^4+...+}^{10}{C_8{x}^8..+}^{14}{C}_{12}{x}^{12}+...{+}^{18}{C}_{16}{x}^{16}+...{+}^{22}{C}_{20}{x}^{20}+...)\times (1+x^4+x^8+x^{12}+x^{16}+x^{20}+...)$
$=1{+}^6{C_4+}^{10}{C_8+}^{14}{C_{12}+}^{18}{C}_{16}{+}^{22}{C}_{20}$
$=1{+}^6{C_2+}^{10}{C_2+}^{14}{C_2+}^{18}{C_2+}^{22}{C}_2$
$=1+\left(\frac{6.5}{1.2}\right)+\left(\frac{10.9}{1.2}\right)+\left(\frac{14.13}{1.2}\right)+\left(\frac{18.17}{1.2}\right)+\left(\frac{22.21}{1.2}\right)$
$=1+15+45+91+153+231=536$