The coefficient of ${x^{53}}$ in the expansion $\mathop \sum \limits_{m = 0}^{100} {\;^{100}}{C_m}{\left( {x - 3} \right)^{100 - m}}{.2^m},\;{\rm{is}}$

JEE MATHEMATICS MCQ: BINOMIAL THEOREM

1 Easy

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Application +4 | -1

The coefficient of

{x^{53}}

in the expansion

\sum \limits_{m = 0}^{100} {\;^{100}}{C_m}{\left( {x - 3} \right)^{100 - m}}{.2^m},\;\mathrm{is}

View answer and explanation

Correct option: C

- {\;^{100}}{C_{53}}

Option C (-^{100}C_{53}) is correct because the given summation simplifies to (x-1)¹⁰⁰, and the coefficient of x⁵³ is -^{100}C_{53}.

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\left({ }^{10} C_0\right)^2-\left({ }^{10} C_1\right)^2+\ldots-\left({ }^{10} C_9\right)^2+\left({ }^{10} C_{10}\right)^2

a

d

\left( {n + 1} \right)

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(1-x)^{201}\left(1+x+x^2\right)^{200}

\left(\frac{1}{x}+2 x\right)^n

{x^5}

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\left(\begin{array}{c}2 n+1 \\ 0\end{array}\right)+\left(\begin{array}{c}2 n+1 \\ 3\end{array}\right)+\left(\begin{array}{c}2 n+1 \\ 6\end{array}\right)+\ldots=170

n