Question

Prove that the coefficient of $x^4$ in the expansion of ${(1+x+x^2+x^3)}^n$ is ${}^n{C}_4{+}^n{C}_2{+}^n{C}_1{\cdot }^n{C}_2$

Answer 1

$(1+x+x^2+x^3{)}^n$

= $(1+x^2+x(1+x^2){)}^n$

= $(1+x^2{)}^n(1+x{)}^n$

coefficient of $x^4$ = coefficient of $x^4$ in $(1+x^2{)}^n\times$ coefficient of $x^0$ in $(1+x{)}^n$ $+$ coefficient of $x^2$ in $(1+x^2{)}^n\times$coefficient of $x^2$ in $(1+x{)}^n$ $+$ coefficient of $x^0$ in $(1+x^2{)}^n\times$ coefficient of $x^4$ in $(1+x{)}^n$

${=}^n{C}_2{.1}^{n-2}.(x^2{)}^2\times 1{+}^n{C}_1{.1}^{n-1}(x^2{)}^1{\times }^n{C}_2{.1}^{n-2}.x^2+1{\times }^n{C}_4{.1}^{n-4}.x^4$

${=}^n{C}_2{+}^n{C}_1{\times }^n{C}_2{+}^n{C}_4$