First slide

Binomial theorem for positive integral Index

Question

The coefficient of x⁵³ in the expansion $\sum_{m = 0}^{100}^{100} C_{m} (x - 3)^{100 - m} 2^{m}$ is

Moderate

A

$^{100} C_{47}$
B

$^{100} C_{53}$
C

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

$^{100} C_{100}$

Solution

The given sigma is the expansion of $[(x - 3) + 2]^{100} = (x - 1)^{100} = (1 - x)^{100}$
Therefore, x⁵³ will occur in T₅₄.
$T_{54} =^{100} C_{53} (- x)^{53}$
Therefore, the coefficient is $-^{100} C_{53}$

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