First slide

Binomial theorem for positive integral Index

Question

The coefficient of X⁵³ in the following expansion
$\sum_{m = 0}^{100}^{100} C_{m} (x - 3)^{100 - m} \cdot 2^{m} is$

Moderate

A

$^{100} C_{47}$
B

$^{100} C_{53}$
C

$-^{100} C_{53}$
D

$-^{100} C_{100}$

Solution

The given sigma expansion
$\sum_{m = 0}^{100}^{100} C_{m} (x - 3)^{100 - m} \cdot 2^{m}$ Can be written as
$\begin{array}{l} {(x - 3 + 2)}^{100} = (x - 1)^{100} = (1 - x)^{100} \\ ∴ Coefficient of x^{53} in (1 - x)^{100} = {(- 1)^{53}}^{100} C_{53} \\ = -^{100} C_{53} \end{array}$

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