First slide

Binomial theorem for positive integral Index

Question

The coefficient of $x^{53}$ in $\sum_{m = 0}^{100} 100 C_{m} (x - 5)^{100 - m} (4)^{m}$ is

Moderate

A

$^{100} C_{47}$
B

$^{100} C_{53}$
C

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

$2^{53}$

Solution

$\sum_{m = 0}^{100}^{100} C_{m} (x - 5)^{100 - m} (4)^{m} = (x - 5 + 4)^{100} = (x - 1)^{100}$
$∴$ Coefficient of $x^{53}$ is $^{100} C_{53} (- 1)^{47} = -^{100} C_{53}$ .

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