In expansion of ${(x^{2} - x - 2)}^{5}$ , the coefficient of $x^{2}$ is

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Solution

$(x - 2)^{5} (1 + x)^{5} = - (2 - x)^{5} (1 + x)^{5}$
= $- [^{5} C_{0} 2^{5} -^{5} C_{1} 2^{4} x +^{5} C_{2} 2^{3} x^{2} + . . . . . . . . . . . .]$ $[^{5} C_{0} +^{5} C_{1} x +^{5} C_{2} x^{2} + . . . . . . . . . . . .]$
$∴$ Coefficient of $x^{2}$ is $-^{5} C_{0} 2^{5} \cdot^{5} C_{2} +^{5} C_{1} \cdot 2^{4} \cdot^{5} C_{1} -^{5} C_{2} 2^{3} = 0$

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