Question

Coefficient of $x^6$ in $\left(\right(1+x\left)\right(1+x^2{)}^2(1+x^3{)}^3....(1+x^n{)}^n)$ is

• A. 26
• B. 28
• C. 30
• D. 35

Answer 1

The correct option is B 28

The coefficient of $x^6$ in the given expression =$\left(\right(1+x\left)\right(1+x^2{)}^2(1+x^3{)}^3....(1+x^6{)}^6)=.(1+x^6{)}^6(1+x^5{)}^5+.......(1+x^2{)}^2(1+x^3{)}^3+(1+x)$
=coefficient of $x^6$ in $\left(1{+}^6{C}_1{x}^6\right)\left(1{+}^5{C}_1{x}^5\right)\left(1{+}^4{C}_1{x}^4\right)\left(1{+}^3{C}_1{x}^3{+}^3{C}_2{x}^6\right)\left(1{+}^2{C}_1{x}^2{+}^2{C}_2{x}^4\right)(1+x)$
= coefficient of $x^{6}in(1+6x^6+5x^5+4x^4)(1+2x^2+3x^3+x^4+6x^5+3x^6)(1+x)$
= coefficient of $x^{6}in(11x^5+17x^6)(1+x)=11x^6+17x^6=28x^6$ is 28.