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# 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}}$  Answer Verified
(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.

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Coefficient of Linear Expansion  Binomial Expansion Formula  Coefficient of Determination  Multiples of 18  Factors of 18  Table of 18  Decimal Expansion of Rational Numbers  Correlation Coefficient  To Determine the Coefficient of Viscosity of a Given Viscous Liquid  Accounting Entries in the Books of the Consignee  