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

JEE MATHEMATICS MCQ: BINOMIAL THEOREM

1 Hard

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Application +4 | -1

The coefficient of

{x^{53}}

in the following expansion

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

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because the given sum simplifies to

(x-1)^{100}

, and the coefficient of

x^{53}

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

Practice All Chapter MCQs →

\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)

a{C_0} - \left( {a + d} \right){C_1} + \left( {a + 2d} \right){C_2} - ... + ...

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

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

{x^5}

\frac{{1 + {x^2}}}{{1 + x}},\left| x \right| < 1,

\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