Question

The product of non-negative integral solution of $x_1+x_2+x_3+4x_4=20$ is

• A. $436$
• B. $536$
• C. $602$
• D. None of thes

Answer 1

The correct option is B $536$

Number of non-negative integral solution of the given expression=coefficient of $x^{20}$ in ${(1-x)}^{-1}{(1-x)}^{-1}{(1-x)}^{-1}{(1-x^4)}^{-1}$= coefficient of $x^{20}$ in ${(1-x)}^{-1}{(1-x^4)}^{-1}$ = coefficient of $x^{20}$ in $(1+{{}^{3}c}_{1}x+{{}^{4}c}_2{x}^2+{{}^{5}c}_3{x}^3+....+{{}^{10}c}_8{x}^8+..+{{}^{14}c}_{12}{x}^{12}...)\times (1+x^4+x^8+x^{16}+...)$=$(1+{{}^{6}c}_4+{{}^{10}c}_8+{{}^{14}c}_{12}+{{}^{18}c}_{16}+{{}^{22}c}_{20})=(1+{{}^{6}c}_2+{{}^{10}c}_2+{{}^{14}c}_2+{{}^{18}c}_2+{{}^{22}c}_2)$

$=1+15+45+91+153+231=536$