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 (x−1)(x−2)(x−3)...(x−18) , the coefficient of x17 is
(a) 684
(b) −171
(c) 171
(a) - 342
Solution
(Hint: The coefficient of x17 is calculated by the addition of the given series. This can be understood as:-If (x−1)(x−2)=x2−3x+2then 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 x17
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 2n(n+1)
Here, we have n=18
Therefore, we get the sum of these 18 terms as
=−218(18+1)
=−218(19)
=−9(19)
=−171
Which is the required coefficient of the x17
Therefore, the required solution is (b) - 171.
Note: In order to solve these types of questions, the students must have an adequate knowledge of the expansion of the series.