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  • The coefficient of  in the expansion of <img alt="\sum_{m=0}^{100}\left({ }^{100} C_m(x-3)^{100-m} \

The coefficient of x^{53} in the expansion of \sum_{m=0}^{100}\left({ }^{100} C_m(x-3)^{100-m} \cdot 2^m\right) is: 

Option: 1

{ }^{100} C_{47}


Option: 2

{ }^{100} C_{53}


Option: 3

-{ }^{100} C_{53}


Option: 4

-{ }^{100} C_{100}


Answers (1)

best_answer

The summation =(x-3+2)^{100}=(x-1)^{100}
Expanding again, we get
\begin{aligned} & (x-1)^{100}={ }^{100} C_0 x^{100}-{ }^{100} C_1 x^{99}+{ }^{100} C_2 x^{98}-\ldots \ldots . \\ \Rightarrow & \quad \text { coefficient of } x^{53}=-{ }^{100} C_{47}=-{ }^{100} C_{53} \end{aligned}

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avinash.dongre

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