Question

The coefficient of $x^{99}$ in the polynomial $(x-1)(x-2)(x-3)...(x-100)$ is

• A. $1$
• B. $-5050$
• C. $5050$
• D. $-1$

Answer 1

The correct option is B $-5050$

If we multiply the brackets and take $x$ from each bracket except one, that is, we take $x$ from $99$ brackets and the constant term from the last bracket we will have $x^{9}9$

So coefficient of $x^{9}9$ will be sum of all the constant terms taken once in this manner.

Coefficient of $x^{99}=-(1+2+3+....+100)=\frac{-100\left(101\right)}{2}=-5050$