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Question

Question: In the expansion of \[(x - 1)(x - 2)(x - 3)...(x - 18)\] , the coefficient of \({x^{17}}\) is \((a...

In the expansion of (x1)(x2)(x3)...(x18)(x - 1)(x - 2)(x - 3)...(x - 18) , the coefficient of x17{x^{17}} is
(a) 684(a){\text{ }}684
(b) 171(b){\text{ }} - 171
(c) 171(c){\text{ }}171
(a) - 342(a){\text{ - 342}}

Explanation

Solution

(Hint: The coefficient of x17{x^{17}} is calculated by the addition of the given series. This can be understood as:-If (x1)(x2)=x23x+2(x - 1)(x - 2) = {x^2} - 3x + 2then coefficient of x=1+(2)=3x = -1 +(- 2) = -3.

In the question, we are given the expansion as
(x1)(x2)(x3)...(x18)(x - 1)(x - 2)(x - 3)...(x - 18)
Here, we can have the maximum power of x=18x = 18
Now, in order to find out the coefficient of x17{x^{17}}
We will add the coefficients of the given expansion
Such that,
=1+(2)+(3)+...(18)= - 1 + ( - 2) + ( - 3) + ...( - 18)
=123...18= - 1 - 2 - 3... - 18
=(1+2+3...+18)= - (1 + 2 + 3... + 18)
Now, we know that the sum of nn terms is equal to n(n+1)2\dfrac{{n(n + 1)}}{2}
Here, we have n=18n = 18
Therefore, we get the sum of these 1818 terms as
=18(18+1)2= - \dfrac{{18(18 + 1)}}{2}
=18(19)2= - \dfrac{{18(19)}}{2}
=9(19)= - 9(19)
=171= - 171
Which is the required coefficient of the x17{x^{17}}
Therefore, the required solution is (b) - 171(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.