Question

The coefficient of $x^7$ in the expansion of $(1-x-x^2+x^3{)}^6.$ is?

• A. 132
• B. 144
• C. -132
• D. -144

Answer 1

The correct option is D -144

$(1-x-x^2+x^3{)}^6=(1-x{)}^6(1-x^2{)}^6=(1{-}^6{C}_{1}x{+}^6{C}_2{x}^2{-}^6{C}_3{x}^3{+}^6{C}_4{x}^4{-}^6{C}_5{x}^5{+}^6{C}_6{x}^6\left)\right(1{-}^6{C}_1{x}^2{+}^6{C}_2{x}^4{-}^6{C}_3{x}^6+\cdots )$
$\Rightarrow$ coefficient of $x^7$ is
${}^6{C}_1{.}^6{C}_3{-}^6{C}_3{.}^6{C}_2{+}^6{C}_5{.}^6{C}_1=6\times 20-20\times 15+6\times 6=-144$