Question

Assertion :(A): Let $P\left(x\right)=(x+1)(x+2)(x+3)\dots (x+11)$, then coefficient of $x^{10}=64$. Reason: (R): Coefficient of $x^{n-1}$ in the product of $(x+1)(x+2)\dots (x+n)$ is $\frac{n(n+1)}{2}$.

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A).
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A) is true but (R) is false.
• D. (A) is false but (R) is true.

Answer 1

The correct option is D (A) is false but (R) is true.

$(x+1)(x+2)(x+3).....(x+n)=$ $=x^n+(1+2+3+....+n)x^{n-1}+\left[\right(1.2+1.3+...+1.n)+(2.3+...+2.n)+...+(n-1)\cdot n]x^{n-2}+...+(1...n)$

$\Rightarrow$ Coefficient of $x^{n-1}=1+2+3+....+n=\frac{n(n+1)}{2}$

$\Rightarrow$ coefficient of $x^{10}=$ coefficient of

$x^{11-1}=$ $=\frac{11\times 12}{2}=66\ne 64$

$\therefore$ Assertion (A) is false but Reason (R) is true.