Find the coefficient of $x^{103}$ in ${(1 + x + x^{2} + x^{3} + x^{4})}^{199} (x - 1)^{201}$

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Solution

Consider the problem

Coefficient of $x^{103}$ in ${(1 + x + x^{2} + x^{3} + x^{4})}^{199} {(x - 1)}^{201}$

Coefficient of $x^{103}$ in

$\begin{matrix} = - {(1 - x)}^{2} {(1 - x^{5})}^{199} = - (1 - 2 x + x^{2}) (1 -^{199} C_{1} x^{5} +^{199} C_{2} x^{10} -^{199} C_{3} x^{15} + . . . . . .) \end{matrix}$

Hence, coefficient of $x^{103} = 0$

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Find the coefficient of x103 in (1+x+x2+x3+x4)199(x−1)201

Consider the problemCoefficient of x103 in (1+x+x2+x3+x4)199(x−1)201 Coefficient of x103 in =−(1−x)2(1−x5)199=−(1−2x+x2)(1−199C1x5+199C2x10−199C3x15+......)Hence, coefficient of x103=0

Find the coefficient of $x^{103}$ in ${(1 + x + x^{2} + x^{3} + x^{4})}^{199} (x - 1)^{201}$