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Greatest Binomial Coefficients

The coefficie...

Question

The coefficient of $x^{53}$ in the expansion of $100 \sum m = 0^{100} C_{m} (x - 3)^{100 - m} 2^{m}$ is:

A

^{100} C_{47}

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B

^{100} C_{53}

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C

-^{100} C_{53}

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D

-^{100} C_{100}

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Solution

The correct option is C $-^{100} C_{53}$
The given sigma is the expansion of
$[(x - 3) + 2]^{100} = (x - 1)^{100} = (1 - x)^{100}$

$∴ x^{53}$ will occur in $T_{54}$ .
$T_{54} =^{100} C_{53} (- x)^{53}$
So, the coefficient is $= -^{100} C_{53}$ .

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The coefficient of $x^{53}$ in the expansion of $100 \sum m = 0$ $^{100} C_{m}$ $(x - 3)^{100 - m} 2^{m}$ is

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\sum_{r = 0}^{100} {^{100} C}_{r} {(x - 3)}^{100 - r} . 2^{r}

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Greatest Binomial Coefficients

Standard XII Mathematics

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