Question

# In the expansion of $(x - 1)(x - 2)(x - 3)...(x - 18)$ , the coefficient of ${x^{17}}$ is$(a){\text{ }}684$(b){\text{ }} - 171$(c){\text{ }}171$$(a){\text{ - 342}}$

(Hint: The coefficient of ${x^{17}}$ is calculated by the addition of the given series. This can be understood as:-If $(x - 1)(x - 2) = {x^2} - 3x + 2$then coefficient of $x = -1 +(- 2) = -3$.

In the question, we are given the expansion as
$(x - 1)(x - 2)(x - 3)...(x - 18)$
Here, we can have the maximum power of $x = 18$
Now, in order to find out the coefficient of ${x^{17}}$
We will add the coefficients of the given expansion
Such that,
$= - 1 + ( - 2) + ( - 3) + ...( - 18)$
$= - 1 - 2 - 3... - 18$
$= - (1 + 2 + 3... + 18)$
Now, we know that the sum of $n$ terms is equal to $\dfrac{{n(n + 1)}}{2}$
Here, we have $n = 18$
Therefore, we get the sum of these $18$ terms as
$= - \dfrac{{18(18 + 1)}}{2}$
$= - \dfrac{{18(19)}}{2}$
$= - 9(19)$
$= - 171$
Which is the required coefficient of the ${x^{17}}$
Therefore, the required solution is $(b){\text{ - 171}}$.

Note: In order to solve these types of questions, the students must have an adequate knowledge of the expansion of the series.