The polynomial P(x)=1−x+x2−x3+−−−−+x16−x17 can be written as a polynomial in y where y=x+1. If the coefficient of y2 be k. Then the value of [k400] is____

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The polynomial $P (x) = 1 - x + x^{2} - x^{3} + - - - - + x^{16} - x^{17}$ can be written as a polynomial in y where $y = x + 1.$ If the coefficient of $y^{2}$ be k. Then the value of $[\frac{k}{400}]$ is_____ (where [ ] represent GIF)

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Detailed Solution

$P (x) = \frac{1 - x^{18}}{1 + x} \Rightarrow C o e f f i c i e n t o f y^{2} i n (\frac{1 - {(y - 1)}^{18}}{y})$ $\Rightarrow$ Coefficient of $y^{3}$ in ${(1 - y)}^{18} =^{18} C_{3} = k \Rightarrow k = 816$