Question

Coefficient of $x^{19}$ in the polynomial $(x–1)(x–2)...(x–20)$ is equal to

• A. $210$
• B. $-210$
• C. $20!$
• D. None of these

Answer 1

The correct option is B

$-210$

Explanation for the correct option:

Step 1: Use the binomial property to get the coefficient:

As the required coefficient is of $x^{19}$.

So, in all the terms, one term will be chosen from the constant each time.

So the coefficient will be $\left(-1\right)+\left(-2\right)+\left(-3\right)......+(-20)$

Step 2: Find the sum to get the coefficient:

Us the summation formulas to get the sum.

$\begin{array}{rcl}s& =& \frac{-20·21}{2}\\ & =& -210\end{array}$ $\left[\because \underset{i=1}{\overset{n}{\sum i}}=\frac{n\left(n+1\right)}{2}\right]$

Hence, option (B) is correct .