Question

The coefficient of $x^8$in the product $(x-1)(x-2)(x-3)...(x-10)$ is

• A. $1024$
• B. $1300$
• C. $1320$
• D. $1360$

Answer 1

The correct option is C

$1320$

Explanation for Correct answer

Given the expression$(x-1)(x-2)(x-3)...(x-10)$

To find: The coefficient of $x^8$

using expansion

$Weknow(x-a)(x-b)=x^2-(a+b)x+ab(x-1)(x-2)$

Rearranging equation gives

$$\begin{gathered} (x-1)(x-2)=x^2-(2+1)x+2\times 1 \\ (x-a_1)(x-a_2)\dots (x-a_n)=x_n–(a_1+a_2+\dots a_n)x^{n-1}+\dots \dots (a_1{a}_2\dots a_{}n) \end{gathered}$$

Coefficient of x⁸$$\begin{gathered} \Rightarrow (2+3+4+..10)+2(3+4+\dots 10)+\dots .9\times 10 \\ =1\times 54+2\times 52+3\times 49+\dots 9\times 10 \\ =1320 \end{gathered}$$

Hence option C is correct